Gauging the Spacetime Code

• The paper’s main contribution is the promotion of the spacetime code to a lattice gauge theory, linking Gauss laws with equivalence classes of spacetime errors.
• It details how elementary circuit operators (ECOs) propagate errors and stabilize measurements, enabling fault diagnosis through causal stabilizer transport.
• Applications include measurement-based quantum computation, topological memory, and learning noise degrees of freedom via gauge-invariant detector observables.

“Gauging the spacetime code” denotes a construction in which the spacetime code is promoted to a lattice gauge theory. In that reformulation, Gauss laws correspond to equivalence relations between configurations of spacetime errors, Wilson loops correspond to detectors, and the resulting theory is used to describe foliated computation, one version of a gauge theory for measurement-based quantum computation, classical memory in a class of topologically ordered mixed states, and the learnable degrees of freedom of circuit Pauli noise (Lee, 4 Jun 2026).

1. Circuit-spacetime kinematics

The underlying setting is a Clifford circuit on $n$ qubits with discrete time $t=0,1,\dots,T$. Between the integer slices $t$ and $t+1$, the circuit places a Clifford operation at half-integer time $t+0.5$,

$C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$

Here $U^{t>}$ is a Clifford unitary and $$\mathcal{M}^{t>}=\{q^{t>}_1,\dots,q^{t>}_{m^{t>}}\}$$ is a set of commuting Pauli measurements (Lee, 4 Jun 2026).

The spacetime discretization distinguishes two types of locations. Data qubits occupy integer-time sites $(i,t)$, with $i=1,\dots,n$, and support the full Pauli algebra through operators such as $t=0,1,\dots,T$0 and $t=0,1,\dots,T$1. Measurement ancilla locations $t=0,1,\dots,T$2 are inserted for each $t=0,1,\dots,T$3, but are treated as classical bits: only a single Pauli, taken to be $t=0,1,\dots,T$4, is allowed there, representing readout error. This organizes the circuit directly as a spacetime object rather than as a sequence of abstract gates (Lee, 4 Jun 2026).

Several projection operations are used repeatedly. For a spacetime Pauli $t=0,1,\dots,T$5, $t=0,1,\dots,T$6 denotes its restriction to time slice $t=0,1,\dots,T$7; $t=0,1,\dots,T$8 and $t=0,1,\dots,T$9 denote its restrictions to integer and half-integer locations, respectively. The time-ordered product of unitaries is written

$t$0

for $t$1. This notation is the basic bookkeeping needed to express error propagation, stabilizer transport, and later gauge-theoretic reinterpretation (Lee, 4 Jun 2026).

2. Elementary circuit operators and stabilizer transport

The central algebraic structure is the set of Elementary Circuit Operators, denoted $t$2. These operators encode how the circuit propagates Paulis and how measurements are registered in spacetime. They come in three families (Lee, 4 Jun 2026).

For each measured Pauli $t$3, the measurement-slice operator is

$t$4

For each qubit $t$5, one introduces propagators for the single-qubit $t$6 and $t$7. If $t$8 is the set of measurement indices $t$9 such that

$t+1$0

then

$t+1$1

Similarly, if $t+1$2 is the set of $t+1$3 for which

$t+1$4

then

$t+1$5

Finally, for each measurement ancilla bit at time $t+1$6, the formal dephaser is

$t+1$7

These commute with the remaining operators and are included to match the subsystem spacetime code formalism (Lee, 4 Jun 2026).

The ECOs determine the causal evolution of stabilizers. Given the noiseless state $t+1$8 at time $t+1$9, the instantaneous stabilizer group $t+0.5$0 is its stabilizer group. A key structural result is

$t+0.5$1

and these generators can be chosen causally, using only ECOs from earlier times. Conversely, if $t+0.5$2 is supported only on a single time slice $t+0.5$3 and generated from earlier ECOs, then $t+0.5$4. The ECOs therefore encode the entire noiseless stabilizer transport of the circuit (Lee, 4 Jun 2026).

3. Faults, equivalence classes, and detectors

A fault is represented as a Pauli configuration on spacetime, combining data faults on integer-time slices with readout faults on half-integer measurement locations. The effect of a fault is separated into two pieces: a shift $t+0.5$5 of the measurement outcomes and a final-time Pauli $t+0.5$6 acting on the output state. The paper terms $t+0.5$7 the bare effect of the fault (Lee, 4 Jun 2026).

The ECOs provide an explicit way to propagate a fault to its effective form. If $t+0.5$8 is the spacetime Pauli representing the fault, then one forms an effective operator $t+0.5$9 by multiplying $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$0 by a time-ordered product of propagator ECOs. The paper states that $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$1 gives the final Pauli $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$2, while $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$3 determines which measurement outcomes are flipped. Multiplying by ECOs does not change the bare effect (Lee, 4 Jun 2026).

This leads to the central equivalence relation. Two faults are equivalent when they induce the same measurement-outcome shift and their final Paulis differ only by an element of $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$4, up to sign. The decisive structural statement is Proposition 2.9: $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$5 if and only if $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$6 and $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$7 are equivalent faults. Thus the ECOs generate exactly the equivalence relation on fault configurations (Lee, 4 Jun 2026).

Detectors are subsets of measurements whose parity is deterministic in every noiseless run, independent of the input state. To each measurement $C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$8, the paper associates a spackle operator,

$C^{t+0.5}=(U^{t>},\mathcal{M}^{t>}).$9

and for multiple measurements the spackle is the product of individual spackles. Proposition 2.13 then states that the spackle of a detector lies in the center of

$U^{t>}$0

At the pre-gauge level, detectors are therefore already singled out as central, circuit-defined observables (Lee, 4 Jun 2026).

4. Gauge-theoretic reinterpretation

The gauging step converts this spacetime-code algebra into a lattice gauge theory. The paper states three defining correspondences: the gauged theory inherits the elements of fault tolerance associated with the circuit, Gauss laws correspond to equivalence relations between configurations of spacetime errors, and Wilson loops correspond to detectors (Lee, 4 Jun 2026).

The earlier circuit-theoretic structures identify the algebraic content of those statements. Proposition 2.9 shows that ECO-generated relations partition faults into equivalence classes with identical bare effect. This suggests the form of the later Gauss-law constraint: physically equivalent spacetime-error configurations are identified by a local gauge relation rather than treated as distinct configurations. Likewise, the centrality of detector spackles in $U^{t>}$1 suggests why the gauged theory can reinterpret detectors as gauge-invariant loop observables. The paper’s abstract makes that correspondence explicit in gauge-theoretic language (Lee, 4 Jun 2026).

A further conceptual shift concerns fault tolerance itself. The work places the spacetime code in contact with the study of dynamical phases by treating fault tolerance as a dynamically stable process rather than only as a static code property. Gauging is the bridge between those viewpoints: circuit-local error equivalences become gauge redundancies, while detector data become Wilson-loop observables. This suggests a reformulation in which decoding data, error transport, and logical protection are expressed directly as gauge-invariant structure on spacetime (Lee, 4 Jun 2026).

5. Applications

The paper identifies three application domains for the gauged theory. First, it contains foliated computation in its description and thereby yields one version of a gauge theory for measurement-based quantum computation. In this usage, the circuit is already organized as a spacetime object with measurements at half-integer slices, so the move to a lattice gauge theory preserves the spacetime-computational organization rather than replacing it (Lee, 4 Jun 2026).

Second, for a class of topologically ordered mixed states, the construction provides a gauge-theoretic language for the classical memory associated with the state. The abstract does not reduce this claim to a particular model in the excerpted details, but the point is precise: the memory structure is described by gauge-theoretic variables rather than by a separate phenomenological parametrization (Lee, 4 Jun 2026).

Third, the gauge-invariant observables of the theory that describe detectors coincide with the learnable degrees of freedom of circuit Pauli noise. This places detector observables simultaneously in three roles: as circuit-defined parity checks, as Wilson-loop observables of the gauged theory, and as the degrees of freedom accessible to learning-theoretic reconstruction of Pauli noise. A plausible implication is that the gauged formulation does not merely repackage decoding data; it isolates exactly the observables that remain both physically invariant and statistically identifiable (Lee, 4 Jun 2026).

6. Relation to other gauging programs

The phrase “gauging the spacetime code” belongs to a wider family of constructions in which gauging acts on data already organized over spacetime rather than only on microscopic Hilbert spaces. In a higher-categorical setting, a global $U^{t>}$2-symmetry of an $U^{t>}$3-dimensional QFT can be encoded by a once-categorified $U^{t>}$4-dimensional SymTFT $U^{t>}$5, and gauging is implemented by replacing the boundary module category of a fixed bulk SymTFT. In that formulation, gauging, symmetry breaking, and adding charged matter become operations on boundaries and module categories, i.e. on “spacetime-code data” (Stockall et al., 11 Nov 2025).

A second related direction is higher gauging of higher-form symmetries on submanifolds. Gauging a discrete $U^{t>}$6-form symmetry on a codimension-$U^{t>}$7 manifold produces condensation defects, often non-invertible, and in $U^{t>}$8 dimensions the fusion “coefficients” are generally $U^{t>}$9d TQFTs rather than numbers (Roumpedakis et al., 2022). Closely related half-spacetime gauging of $$\mathcal{M}^{t>}=\{q^{t>}_1,\dots,q^{t>}_{m^{t>}}\}$$0 in $$\mathcal{M}^{t>}=\{q^{t>}_1,\dots,q^{t>}_{m^{t>}}\}$$1d produces non-invertible duality surfaces obeying a fusion 2-category and admits a $$\mathcal{M}^{t>}=\{q^{t>}_1,\dots,q^{t>}_{m^{t>}}\}$$2d BF-theory SymTFT description on a slab (Cui et al., 2024).

Taken together, these programs use different technical languages—Clifford-circuit subsystem spacetime codes, higher categories of quasi-coherent sheaves, and defect-based higher-form gauging—but they share a structural move: gauging is applied to spacetime-organized algebraic data. In (Lee, 4 Jun 2026), that data is the circuit’s ECO-generated error algebra; in (Stockall et al., 11 Nov 2025), it is the SymTFT and its boundary modules; in (Roumpedakis et al., 2022) and (Cui et al., 2024), it is the defect network of higher-form symmetries. This suggests that “gauging the spacetime code” names a general strategy in which local equivalence relations, conserved detector data, and topological observables are recast as gauge structure on spacetime.