A Bravyi-König theorem for Floquet codes generated by locally conjugate instantaneous stabiliser groups

The Bravyi-König (BK) theorem is an important no-go theorem for the dynamics of topological stabiliser quantum error correcting codes. It states that any logical operation on a $D$-dimensional topological stabiliser code that can be implemented by a short-depth circuit acts on the codespace as an element of the $D$-th level of the Clifford hierarchy. In recent years, a new type of quantum error correcting codes based on Pauli stabilisers, dubbed Floquet codes, has been introduced. In Floquet codes, syndrome measurements are arranged such that they dynamically generate a codespace at each time step. Here, we show that the BK theorem holds for a definition of Floquet codes based on locally conjugate stabiliser groups. Moreover, we introduce and define a class of generalised unitaries in Floquet codes that need not preserve the codespace at each time step, but that combined with the measurements constitute a valid logical operation. We derive a canonical form of these generalised unitaries and show that the BK theorem holds for them too.

💡 Research Summary

In this work the authors extend the celebrated Bravyi‑König (BK) no‑go theorem to the setting of Floquet quantum error‑correcting codes, a class of dynamical codes in which the stabiliser group changes from one time step to the next. The original BK theorem states that any logical operation on a D‑dimensional topological stabiliser code (TSC) that can be implemented by a constant‑depth, geometrically local circuit must belong to the D‑th level of the Clifford hierarchy. The authors first formalise a precise definition of Floquet codes based on locally conjugate instantaneous stabiliser groups (ISGs). An ISG sequence A₁ → A₂ → … → A_τ is required to consist of reversible pairs: two stabiliser groups A and B are reversible (or conjugate) if there exist generating sets such that each generator of A anticommutes with exactly one generator of B and commutes with all the others, and vice‑versa. This structure guarantees that measuring the generators of B (or projecting onto the joint eigenspace of B) does not reveal any logical information – the measurement outcomes are uniformly random – yet the logical subspace is preserved up to a unitary equivalence.

The paper proceeds in two stages. In the first stage the authors consider code‑preserving unitaries, i.e. at each time step a unitary U_t that maps the current code space onto itself. Because the transition from A_t to A_{t+1} can be represented by a constant‑depth unitary K_{A_t,A_{t+1}} (derived from the projection Π_{A_{t+1}}), the overall logical operation is a product of alternating K‑maps and the U_t’s. This situation falls directly under the original BK theorem, and therefore any logical gate realizable in this way is confined to the D‑th Clifford level.

The novel contribution lies in the second stage, where the authors introduce generalised logical unitaries. These are unitaries that need not keep the system inside the instantaneous code space at the moment they are applied; they are allowed to temporarily leave the code space provided two conditions hold after the subsequent measurement: (i) all pre‑existing detectable errors remain detectable and all self‑correctable errors remain self‑correctable (error‑detectability and self‑correction condition); (ii) logical information is unchanged and the effective logical action is independent of the specific measurement outcomes (logical preservation and equivalence condition). The authors formalise these conditions in Section V and prove that any unitary satisfying them can be brought to a canonical form
 Ũ_t = K_{A_t,A_{t+1}} · V_t · K_{A_t,A_{t+1}}†,
where V_t is a geometrically local, constant‑depth unitary that respects the same locality constraints as the original circuit. The K‑operators are the unitary representatives of the ISG transition (essentially the square‑root of the projection onto the next stabiliser group). This decomposition shows that the “non‑code‑preserving” part of the operation can be absorbed into the ISG transition, leaving a purely local unitary V_t that is subject to the BK bound.

The central result, Theorem 1, states that for a finite Floquet sequence of length τ = O(1) in which each step employs a constant‑depth, constant‑range generalised logical unitary, the total logical transformation implemented by the whole sequence (unitaries plus measurements) still lies in the D‑th level of the Clifford hierarchy. The proof follows the same logic as the original BK theorem: after replacing each measurement‑induced projection by its unitary K‑equivalent, the entire process becomes a constant‑depth circuit acting on a D‑dimensional TSC, and thus the hierarchy bound applies.

Beyond the formal theorem, the paper offers several insights of broader relevance. First, it clarifies the mechanism by which Floquet codes preserve logical information despite apparently destructive projective measurements: the anti‑commutation structure of reversible pairs ensures that the projection is proportional to a unitary, and logical operators are either unchanged or map to equivalent representatives in the next stabiliser group. Second, the introduction of generalised logical unitaries expands the toolbox for designing logical gates in dynamical codes, allowing one to exploit measurement‑induced transitions without violating fault‑tolerance. Third, the notion of local conjugacy emerges as a unifying structural property that simultaneously guarantees bounded error propagation and the applicability of the BK hierarchy bound.

In the discussion, the authors note that while their analysis focuses on locally conjugate ISGs (which cover most known Floquet constructions such as the honeycomb code and its variants), the techniques should extend to broader classes of dynamical codes, possibly with minor modifications. They also hint at future directions, including the systematic synthesis of logical gate sets that saturate the BK bound in Floquet settings, and the exploration of higher‑dimensional Floquet codes where the D‑th‑level hierarchy becomes a more stringent constraint.

Overall, the paper successfully bridges a gap between static topological stabiliser codes and their dynamical Floquet counterparts, demonstrating that the powerful BK no‑go theorem remains valid even when the code space is allowed to evolve, provided the evolution respects local conjugacy and the generalized logical unitary conditions. This result strengthens the theoretical foundations of Floquet codes and paves the way for fault‑tolerant logical gate design in time‑dependent quantum error‑correction architectures.