Planar Code States in Quantum Error Correction

• Planar code states are stabilizer states defined on planar lattices with local X and Z checks, encoding logical qubits via boundary-supported string operators.
• They encompass both traditional surface-code constructions and planar maximally entangled (PME) states, enabling fault-tolerant encoding and logical operations through lattice surgery techniques.
• Their design is pivotal for diverse architectures, influencing decoder performance, resource overhead, and implementations from rotated qubit patches to bosonic mode realizations.

Searching arXiv for recent and foundational papers on planar code states, surface/planar codes, and related “planar maximally entangled” states. Planar code states usually denote the stabilizer states of finite surface-code patches: simultaneous $+1$ eigenstates of local $X$-type and $Z$-type checks on a planar lattice, with logical information stored in nontrivial strings between boundaries. In a distinct but related entanglement-theoretic usage, the term is also applied to “planar maximally entangled” states, where maximal mixedness is required only for connected regions compatible with an underlying planar geometry rather than for arbitrary subsystems. Across these usages, the topic spans topological stabilizer codes, graph-state and MBQC representations, lattice-surgery dynamics, decoding under structured noise, and geometry-constrained multipartite entanglement (Horsman et al., 2011, Doroudiani et al., 2020).

1. Terminology and scope

In the surface-code literature, a planar code state is any state in the code space of a planar patch. For a stabilizer group $\mathcal{S}$ generated by face and vertex operators, the code space is

$$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$

equivalently the ground space of a planar surface-code Hamiltonian. For the standard planar patch discussed in the lattice-surgery literature, $\dim\mathcal{C}=2$, so one patch encodes one logical qubit (Horsman et al., 2011).

A separate literature uses planar geometry in a different sense. There, “planar code states” appear as Planar Maximally Entangled (PME) states on a 1D ring of qudits, defined by the requirement that every connected subset of size at most half the system has maximally mixed reduced state. The original PME work is explicit that these are not “codes” in the strict stabilizer or $[[n,k,d]]$ sense, although they are code-like many-body states with geometry-constrained entanglement and secret-sharing behavior (Doroudiani et al., 2020).

This terminological bifurcation matters. The stabilizer-code usage refers to topological code spaces with local parity checks and logical strings. The PME usage refers to multipartite entangled pure states whose information-theoretic properties are restricted to connected regions. The two traditions intersect conceptually through geometry-respecting entanglement, but they are not identical constructions.

2. Stabilizer structure of planar surface-code states

The standard planar surface code is a CSS stabilizer code on a finite square lattice with open boundaries. The literature represented here uses both edge-based and vertex-based conventions for physical qubits, but in the canonical surface-code formulation with qubits on edges the stabilizer generators are

$$A_v=\prod_{e\ni v}X_e,\qquad B_p=\prod_{e\in\partial p}Z_e,$$

where $A_v$ is a site or star operator and $B_p$ a plaquette operator. Boundary terms have reduced weight, but remain products of $X$0 around vertices and $X$1 around faces (Higgott et al., 2020, Horsman et al., 2011).

Planar boundaries split into rough and smooth types. Rough boundaries condense $X$2-type excitations and support logical $X$3 strings connecting the two rough sides; smooth boundaries condense $X$4-type excitations and support logical $X$5 strings connecting the two smooth sides. The code distance $X$6 is the minimum weight of a nontrivial logical operator, so for the usual $X$7 planar geometry one has $X$8 (Horsman et al., 2011, Higgott et al., 2020).

The resource overhead depends strongly on layout. In the standard planar encoding the number of data qubits is $X$9, whereas the rotated planar encoding uses $Z$0 data qubits while preserving distance. Concrete examples given in the lattice-surgery work are 13 data qubits for a distance-3 standard planar patch, 9 data qubits for a distance-3 rotated patch, and 25 data qubits for a rotated distance-5 patch; with four reused syndrome qubits, the rotated distance-3 planar code can be implemented with 13 total qubits (Horsman et al., 2011).

The planar code state can also be viewed as the ground space of

$Z$1

This Hamiltonian viewpoint is important because it makes clear that planar code states are simultaneously stabilizer states, topologically ordered states on finite patches, and encoded logical states distinguished by boundary-supported string operators (Horsman et al., 2011, Higgott et al., 2020).

3. Encoding, surgery, and logical-state manipulation

A central operational question is how to prepare an unknown logical state $Z$2 by local unitaries. For the distance-$Z$3 planar code, an optimal local-unitary encoder with exactly $Z$4 time steps is known. It uses only nearest-neighbor CNOT and Hadamard gates, starts from one unknown physical qubit plus product-state ancillas, and grows the code by an $Z$5 inductive step costing four time steps. This saturates the linear lower bound for local-unitary encoding of the surface code (Higgott et al., 2020).

Once prepared, planar code states are manipulated fault-tolerantly by lattice surgery. A rough merge initializes intermediate data qubits in $Z$6, introduces new $Z$7-type checks across the boundary, and effects a measurement of $Z$8; a smooth merge initializes in $Z$9 and measures $\mathcal{S}$0. The inverse operations are splits. For a smooth split,

$\mathcal{S}$1

so the original logical $\mathcal{S}$2 eigenvalue is duplicated while logical $\mathcal{S}$3 becomes nonlocal across the two daughter patches. These merge and split primitives are nonunitary at the logical level but fault-tolerant under repeated syndrome extraction (Horsman et al., 2011).

Logical CNOT follows from a smooth merge between control and an intermediate patch, a smooth split, and then a rough merge between the intermediate patch and the target. The same framework generates encoded Bell and GHZ states by repeated smooth or rough splits. The lattice-surgery paper also gives concrete small-scale resource counts: a distance-3 logical CNOT between two rotated planar qubits plus intermediate space can be implemented with 33 data qubits and 20 syndrome qubits, for a total of 53 physical qubits (Horsman et al., 2011).

These constructions establish a characteristic operational picture of planar code states: preparation by local Clifford circuits, topological storage via boundary-defined logical operators, and logical processing by surgery rather than braiding or transversal long-range gates.

4. Decoding, biased-noise adaptations, and dynamical planar variants

Planar code states are only useful as protected resources when accompanied by decoders matched to the syndrome geometry. A simple hierarchical decoder for the planar code pairs anyons or boundaries in increasing Manhattan shells. In the noiseless-syndrome setting its worst-case complexity is

$\mathcal{S}$4

and Monte Carlo benchmarks give a bit-flip threshold of approximately $\mathcal{S}$5. Under nearest-neighbor correlated noise with correlation parameter $\mathcal{S}$6, the threshold shifts to approximately $\mathcal{S}$7, consistent with the expected reduction from the increased effective error fraction (Wootton, 2013).

A more recent non-CSS modification is the XYZ planar code. Here the usual $\mathcal{S}$8-type stabilizers are retained, but each $\mathcal{S}$9-type plaquette stabilizer is deformed into a mixed $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$0 operator, producing a non-CSS stabilizer group on the same planar geometry with $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$1 data qubits. The associated posterior MWPM decoder reweights edges using local posterior probabilities inferred from neighboring $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$2-syndrome bits, while keeping essentially the same blossom-dominated asymptotic complexity $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$3. Reported thresholds include $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$4 and $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$5, and the abstract reports an improvement of about $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$6 over the standard planar code in the infinite-bias limit (Wang et al., 22 May 2026).

A different departure from static stabilizer structure is the planar Floquet code. In that setting, the instantaneous stabilizer group rotates under a three-step measurement schedule on a planar color-code-like lattice with boundaries. At each time slice the code is locally equivalent to a planar homological code with alternating rough and smooth boundaries, and the rotating schedule implements a logical Hadamard after three steps. However, the planar boundary construction analyzed there has constant space-time distance and therefore is not fault-tolerant as a planar quantum memory (Vuillot, 2021).

Together these results show that planar code states are not a single fixed object but a family of geometry-constrained logical states whose robustness depends on the decoder, on the noise bias, and on whether the stabilizer structure is static or Floquet-driven.

5. Graph-state, toric, and MBQC representations

Every stabilizer state is local-Clifford equivalent to a graph state, and planar code states admit especially structured graph-theoretic descriptions. For the toric code, an explicit LC-equivalent graph state can be written whose graph decomposes into star graphs and half graphs. The star graphs are LC-equivalent to GHZ states, while the half graphs couple these GHZ-like components so that the resulting stabilizer state has local toric-code checks. This representation yields a log-depth preparation circuit assuming geometrically non-local gates, and a constant-depth construction with ancillae and measurements at the cost of increased circuit width (Liao et al., 2021).

A more abstract graph-theoretic classification identifies bipartite circle graph states with planar code states, up to local Clifford equivalence. In that formulation, a planar code state is constructed from a planar multigraph with one qubit per edge, $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$7-type star operators on vertices, and $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$8-type plaquette operators on faces; conversely, every bipartite circle graph state is LC-equivalent to such a planar code state. This correspondence implies two notable consequences: first, planar code states satisfy $$\mathcal{C}=\{\,|\psi\rangle: S|\psi\rangle=|\psi\rangle\ \forall S\in\mathcal{S}\,\},$$9; second, MBQC on planar code states is efficiently classically simulable, and the same efficient simulability extends to all circle graph states (Hahn et al., 9 Mar 2026).

A complementary entanglement-focused representation rewrites toric-code states on arbitrary planar graphs, embedded on the torus, into tensor products of Kitaev ladder states via non-local disentanglers along non-contractible cycles satisfying a topological constraint. In that picture, the short-range entanglement is confined to the ladders, while the universal long-range entanglement of the toric-code family is encoded in the non-local pattern relating the ladders (Zarei et al., 3 Sep 2025).

These graph-state and non-local representations sharpen the structural status of planar code states. They are simultaneously topological stabilizer states, graph-state objects with rigid local-equivalence classes, and MBQC resources whose entanglement is substantial but not computationally universal.

6. Architecture-level realizations: bosonic and high-density planar code states

Planar code states have also been embedded into hardware-specific constructions. One route concatenates binomial rotation-symmetric bosonic codes with a planar outer code in a cluster-state MBQC scheme. In this architecture, each planar-code qubit is itself a binomial code hosted by a single oscillator, and planar-code stabilizers are realized by logical $\dim\mathcal{C}=2$0-basis measurements derived from bosonic phase measurements. The study concludes that good performance requires adaptive phase measurements, maximum-likelihood qubit-state inference, and weighted MWPM decoding; with these ingredients, thresholds up to $\dim\mathcal{C}=2$1 are reported for some binomial-code parameter choices (Soule et al., 2023).

Another route increases logical density while remaining strictly planar. A 2026 construction on a regular 2D hex grid introduces a denser planar surface code with twist defects, six-layer measurement cycles, and padding-free lattice surgery within an optimal $\dim\mathcal{C}=2$2 data-and-measurement-qubit bounding box per patch. Its abstract reports an estimated encoding rate up to $\dim\mathcal{C}=2$3 that of a rotated surface-code patch under a one- and two-qubit $\dim\mathcal{C}=2$4 uniform depolarizing model. The same work reports a $\dim\mathcal{C}=2$5 space improvement and a $\dim\mathcal{C}=2$6 spacetime improvement over prior minimum-Toffoli resource estimates for one utility-scale chemistry application, and gives an example of FeMoco-based phase estimation in under a month with $\dim\mathcal{C}=2$7k noisy superconducting qubits under its stated assumptions (Low et al., 28 May 2026).

These implementations underscore that planar code states are not only abstract code-space vectors. They are practical logical states for architectures ranging from conventional qubit lattices to bosonic modes and high-density twist-defect layouts, with the geometry of the code state directly constraining measurement schedules, decoder design, and resource estimates.

7. PME states as planar code-like entanglement states

In the PME literature, a pure state $\dim\mathcal{C}=2$8 of $\dim\mathcal{C}=2$9 qudits on a 1D ring is called Planar Maximally Entangled if every connected subset $[[n,k,d]]$0 of size $[[n,k,d]]$1 has reduced state

$[[n,k,d]]$2

For even $[[n,k,d]]$3, this is often emphasized at the half-system level: every connected block of $[[n,k,d]]$4 adjacent qudits is maximally mixed. Every AME state is therefore a PME state, but not conversely; the class is strictly larger because disconnected subsystems need not be maximally mixed (Doroudiani et al., 2020).

This relaxation has concrete consequences. No AME state of four qubits exists, yet the PME paper exhibits two inequivalent five-parameter families of four-qubit PME states and explicit PME constructions for any even number of qudits and any local dimension. A canonical family for $[[n,k,d]]$5 is

$[[n,k,d]]$6

a product of Bell pairs arranged so that every connected half-system cut intersects each pair exactly once, yielding a maximally mixed reduced state on the connected half (Doroudiani et al., 2020).

A later construction derives PME states from separable phase states. The phase states are tensor products of single-qudit Fourier eigenstates and therefore separable, but after application of an Ising-type controlled-phase unitary they become graph-state-like entangled states. The paper gives explicit constructions for $[[n,k,d]]$7 and general even $[[n,k,d]]$8, and develops a quantum secret-sharing protocol in which a dealer performs a generalized Bell measurement and authorized adjacent sets reconstruct the secret through stabilizer-like correction operators (Bouhouch et al., 2024).

PME states therefore occupy a distinct position relative to planar surface-code states. They are not stabilizer code spaces with $[[n,k,d]]$9 parameters, but they are planar code-like resource states: geometry-respecting, maximally entangled on connected regions, and directly applicable to teleportation, secret sharing, and error-correction-motivated protocols (Doroudiani et al., 2020, Bouhouch et al., 2024).