Improved QLDPC Surgery: Logical Measurements and Bridging Codes

1 Introduction

• Θ(w) ancilla qubits suffice for measuring a weight w logical operator when its induced Tanner graph is sufficiently expanding.Gauge fixing removes gauge qubits while preserving merged-code distance at least d.Source: §1 · p. 4
• The scheme introduces distance-preserving bridge constructions for Y and joint measurements, enabling fault-tolerant connections between arbitrary QLDPC code blocks.This supports architectures combining multiple QLDPC code families.Source: §1 · p. 4
• d rounds of syndrome extraction ensure fault distance d for the proposed measurement schemes, including the code adapter.The analysis distinguishes code, fault, and circuit distance as separate protection metrics.Source: §1 · pp. 3–4
• A modular decoder decomposes the merged-code space-time graph into original-code and ancilla components and achieves fault distance d.The practical decoder combines BPOSD with matching and is reported to decode substantially faster than fully BPOSD at comparable logical error rates.Source: §1 · p. 4
• The main theoretical limitation is that gauge-fixed operator weights lack general upper bounds, so gauge fixing may break the merged code’s LDPC property.The authors report average-case evidence but no worst-case guarantee; later works are described as resolving these caveats.Source: §1 · p. 4
• With 103 additional qubits, a mono-layer ancilla system fault-tolerantly measures eight distinct logical Pauli operators, each verified to have fault distance 12.The connectivity graph has exactly two degree-8 qubits and all others have degree at most 7.Source: §1 · p. 4
• For the gross code, circuit-level simulations find logical measurement error rates within 10x for X and 5x for Z of the base-code rate, with seven syndrome repetitions optimal at physical error rate 0.001.Combining measurements with permutation automorphisms implements 288 native Pauli measurements and 95 Pauli rotations generating the eleven-qubit Clifford group.Source: §1 · p. 5

2.1 Tanner graph notation

• Tanner graphs represent qubits and X- or Z-checks as vertex sets connected by sparse incidence matrices.Rows denote checks, columns denote qubits, and edge labels specify the matrix and Pauli type.Source: §2.1 · p. 6

2.2 The CKBB ancilla system

• The induced Tanner graph of an irreducible logical X operator contains its support V0 and neighboring Z-checks C0.The same construction applies to Z operators after swapping X and Z.Source: §2.2 · p. 7
• CKBB appends 2d−1 alternating layers connected through repeated incidence matrices, transposes, and identity links.The measured logical operator becomes a product of appended X-checks, enabling code switching for fault-tolerant measurement.Source: §2.2 · p. 7
• The CKBB construction requires more than 2d2 ancilla qubits and can exceed the original code size when d scales as √n.Its time overhead is d rounds of error correction, while maximum degree increases from δ to δ + 1.Source: §2.2 · p. 8
• CKBB preserves code distance d for the subsystem containing all unmeasured logical qubits, but its fault or circuit distance is not analyzed in the reviewed description.Gauge qubits carry no logical information.Source: §2.2 · p. 8

3.1 Gauge-fixed X Ancilla Systems

• The gauge-fixed ancilla system GX adds null(F)-spanning Z checks, promotes gauge degrees of freedom to stabilizers, and makes the measured X operator a stabilizer while removing one logical qubit.A matrix G with GF = 0 supplies the additional checks; minimizing its row and column weights minimizes Tanner-graph degree increases.Source: §3.1 · p. 8
• Without structural knowledge of the code or measured operator, the gauge-check matrix G has no general row- or column-weight bounds, though average-case checks may preserve LDPC structure.Some gauge checks may be products of other Z stabilizers and therefore need not be explicitly measured.Source: §3.1 · p. 9

3.2 X Measurement Protocol

• The X-measurement protocol alternates initialization, fusion, repeated GX-check measurements, and splitting, using deterministic detector parities while extracting the product of unreliable HX(V1) checks.The module is a reusable stabilizer resource state, and R ≥ d is sufficient for fault tolerance, although smaller R may improve practical trade-offs.Source: §3.2 · pp. 9, 11

3.3 Code Distance and Fault Distance

• With ⌈L/2⌉≥1/β, the gauge-fixed merged code GX has distance at least d, and its detector graph has fault distance d for a suitable layer choice.The construction’s space overhead is O(d/β); when induced checks act on exactly two measured-support qubits, expansion can yield linear overhead.Source: §3.3 · pp. 11, 14
• The distance proof separates Z and X logical operators: Z operators retain weight at least d, while expansion prevents X operators from being reduced below d.The expansion condition depends on the boundary Cheeger constant β of the induced Tanner graph; practical instances may require fewer layers than the worst-case bound.Source: §3.3 · pp. 11, 13, 14
• The X system has measurement fault distance min(R, d(¯Z_M L U)) and logical fault distance d(L* U), with U determining these distances as the weakest code.The logical distance is exact, while the measurement distance is exact because faults can occur between the initial measurement rounds.Source: §3.3 · pp. 15–16
• The fault analysis classifies measurement faults as elements of ¯Z_M L U and logical faults as elements of L* U in the shared-detector case, while merged-code detectors preclude measurement faults in the intermediate case.The three detector-group cases correspond to S ∩ S_X, S_X, and S, producing the bounds used in the distance results.Source: §3.3 · pp. 15–16
• The same lemma applies to arbitrary ancilla measurement schemes by replacing ¯Z_M with an anticommuting Pauli ¯Q_M and using a common logical group L for the unaffected qubits.For measuring ¯P_M = ¯X_1 ¯X_2, one may choose ¯Q_M = ¯Z_1 and bound distances using d(¯Q_M L U) and d(L* U).Source: §3.3 · p. 16
• For CSS X-operator measurements, d(L* U) = min(dZ(L* S), dX(L* S_X)) and d(¯Z_M L U) = dZ(¯Z_M L S).The Z-type operators of U match those of S, while its X-type operators match those of S_X.Source: §3.3 · pp. 16–17
• If the merged-code stabilizers are measured at least d times and its X distance is at least d, both measurement and logical fault distances of the X system are at least d.Here d is the distance of the original code; the X-distance condition is satisfied, for example, under Theorem 6.Source: §3.3 · p. 17

Limitations and remedies

• Without additional assumptions on the induced Tanner graph F, the scheme has no worst-case guarantee that null(F) admits a low-weight basis.Consequently, the weights of gauge checks in UL may be unbounded in the worst case; redundant checks and cellulation are proposed remedies.Source: §3.4 · pp. 17, 19

Redundancy condition

• Redundant gauge checks can be omitted from the merged code when an equivalent product of existing Z-checks reproduces the same operator.Lemma 12 gives this condition for c ∈ C0 with cF = 0: if some c̄ ∈ CZ0 satisfies HZ(c̄) = HZ(c), then Z(c ∈ C1) is redundant.Source: §3.4 · p. 17

Practical implication

• Stabilizer redundancy in the original code may allow many or all ancilla gauge checks to be omitted, especially for hypergraph product and lifted product codes.The paper identifies structured redundancy in these code families as a practical heuristic and demonstrates it for hypergraph product codes.Source: §3.4 · p. 17

Hypergraph product example

• For a hypergraph product code built from H with rank r, the construction uses n^2 + m^2 physical qubits and has logical dimension (n − r)^2 + (m − r)^2.The analysis specifies bases of ker H and ker H⊤ to describe the X-logical operators used in the gauge-fixed ancilla construction.Source: §3.4 · pp. 17–18

Redundancy condition

• The construction proves redundancy for gauge checks induced by restrictions of vectors in null(H), because the corresponding Z-checks remain unchanged by the ancilla attachment.The unchanged checks allow Z(c ∈ C1) to be expressed as a product of Z-stabilizers in CZ.Source: §3.4 · pp. 17–18

Limitations and remedies

• An earlier claim that every gauge check from null(F) was redundant was corrected; the proven redundancy applies only under the stated equivalent-check condition.The paper explicitly attributes the correction to an error in the earlier proof.Source: §3.4 · pp. 17, 19

Decoder construction

• The decoder first processes DG, propagates corrections to Danc, and then decodes the remaining ancilla errors separately.This decomposition is the modular structure of the proposed algorithm.Source: §3.5 · p. 19

Ancilla decoding

• Parity matching on a length-R chain can fail only for errors of weight at least R/2, which is at least d/2 when R ≥ d.Each chain node represents a measurement step, and only total detector parity matters for the measurement outcome.Source: §3.5 · p. 19

Decoder guarantee

• Fault distance d is achieved when the original-code decoder corrects errors below d/2, yielding a whole-system correction that triggers no detectors and commutes with the measured observable.The construction assumes a decoder for DG with this guarantee and extends it to D.Source: §3.5 · p. 19
• The matching problem is the only decoding component that can flip the measurement outcome, because the other correction steps preserve the measured observable.The proof reduces potentially harmful undetected errors to chains spanning the measurement sequence.Source: §3.5 · p. 20

Limitation

• The simple line-matching decoder has no threshold under stochastic noise because each step is odd with probability approaching 1/2, causing failure probability approaching 1/2.The authors suggest minimum-weight decoders for both detector-graph components as a possible remedy.Source: §3.5 · p. 20

Implementation

• For the J144, 12, 12K case study, the required adjacency condition holds for selected logical operators, enabling BPOSD on DG and MWPM on Danc.This is the decoder configuration used in the numerical simulations.Source: §3.5 · p. 21

Construction

• Bridge qubits replace CKBB’s 2d −1 layers for joint measurement, with min(|X̄1|, |X̄2|) bridge qubits connected to checks in both ancilla systems.The construction targets joint operators with nonoverlapping supports; overlapping operators use the X-ancilla construction instead.Source: §3.6 · p. 21

Gauge checks

• A complete independent bridge-gauge basis contains |B|−1 checks, with a basis in which each check acts on exactly two bridge qubits.The bridge-restricted checks form adjacent-pair patterns on the bridge qubits.Source: §3.6 · pp. 21, 23

Limitation

• The bridge-gauge basis depends on the measured logical operators and may be non-LDPC in the worst case, although the gross-code study found a basis of weight at most 5.The construction does not optimize these gauge-check weights in general.Source: §3.6 · p. 23

Code parameters

• The merged code has k −1 logical qubits because the bridge adds |B| qubits and |B|−1 independent gauge checks to the two measurement systems.This follows from the independent gauge-check basis for the bridge.Source: §3.6 · pp. 21, 23

Implementation

• The joint protocol can initialize selected ancilla and bridge qubits in |0⟩, measure the merged stabilizers for R rounds, and finish with single-qubit Z measurements.This avoids separately preparing and preserving each module in a stabilizer state.Source: §3.6 · p. 23

Fault distance

• If both ancilla systems satisfy the stated layer-expansion conditions and the merged code is measured for R ≥ d rounds, measurement and logical fault distances are at least d.The conditions use the layer counts and boundary Cheeger constants of the two systems.Source: §3.6 · p. 23

Construction

• The Y construction joins the X and Z systems on the first layer using bridge qubits and a mixed-type check q1.The construction assumes that X̄ and Z̄ overlap on exactly one qubit for the proofs.Source: §3.7 · pp. 24–25

Gauge checks

• The Y-system gauge basis contains |B| independent checks, with a choice having exactly one check acting on each bridge qubit.These additional checks account for bridge-supported gauge degrees of freedom.Source: §3.7 · p. 25

Limitation

• For overlap on more than one qubit, additional gauge checks may appear and must be fixed to retain exactly k −1 encoded qubits.The one-qubit-overlap assumption is used to simplify the gauge-check analysis.Source: §3.7 · pp. 25–26

Code parameters

• The Y-system encodes exactly k −1 qubits.The qubit-count result follows from the structure of the X, Z, and bridge gauge checks.Source: §3.7 · p. 26

Fault distance

• If the X- and Z-systems individually have code distance at least d, the Y-system has logical fault distance at least d.The proof decomposes a nontrivial Y logical operator into X and Z components and shows one component retains weight at least d.Source: §3.7 · p. 26
• With sufficient layers relative to the boundary Cheeger constants and R ≥ d measurement rounds, the Y construction has measurement fault distance at least d.The required conditions are ⌈LX/2⌉≥1/βX and ⌈LZ/2⌉≥1/βZ.Source: §3.7 · p. 27

Fault-tolerant connection

• For separate code blocks with individually fault-tolerant X ancillas and minimal-weight logical representatives, the bridge preserves measurement and logical fault distances at least d when R ≥ d.The same guarantee applies when the two logical operators have expanding induced Tanner graphs and sufficiently many layers.Source: §3.8 · p. 27

Code adaptation

• The bridge can connect arbitrary code blocks for fault-tolerant logical-information teleportation and combine code families with different optimized capabilities.The paper specifically describes connecting a QLDPC code to a distinct magic-state-factory code.Source: §3.8 · pp. 27–28

Limitation

• The bridge construction’s gauge-check weights depend on the measured logical operators and may be high in the worst case.Choosing low-weight bridge checks is formulated as a matrix problem involving the bridge repetition-code check space.Source: §3.8 · p. 28
• For arbitrary Tanner graphs, suitable connecting matrices may not exist, so the basic bridge can fail to preserve the LDPC property.The authors expect reasonable choices in average cases, while identical induced graphs permit a direct submatrix choice.Source: §3.8 · p. 28

Related development

• The LDPC limitation is resolved for simple expander graphs by [SJOY24], which provides an efficiently constructible LDPC bridge for arbitrary QLDPC code families.This extends the basic bridge into a universal code-adapter construction.Source: §3.8 · p. 28

4 Case Study: Gross Code

• The ancilla system adds 103 physical qubits—54 data and 49 check—for various logical measurements, enabling all logical Clifford gates and Pauli measurements on eleven of twelve logical qubits, versus [BCG+24]’s 1380-qubit system.Source: §4 · p. 29

4.1 Mono-layer ancilla system

• The mono-layer construction preserves distance 12 for the individual X and Z merged codes and, with bridge connections, measures X, Y, Z, X′, Y′, XX′, and ZZ′.The same connected-qubit graph realizes these systems, while numerical CPLEX checks confirm distance preservation for the Y, Y′, XX′, and ZZ′ merged codes.Source: §4.1 · pp. 30–31

4.2 Numerical simulations

• The syndrome circuits retain circuit-level distance 10, while logical error rates are 10x and 5x higher than the bare gross-code cycle for the X and Z systems, respectively.The merged syndrome circuits have the same depth per cycle as the gross-code circuit by using an idle-data gap for ancilla connections.Source: §4.2 · pp. 32, 34
• At physical error rate 0.001, measurement and remaining-logical error rates balance after about seven rounds, while modular decoding has similar accuracy to BPOSD but matching is over 100x faster.The merged-code BPOSD decoder is about 10x slower than BPOSD for the gross code alone.Source: §4.2 · p. 34

4.3 Implementing logical Clifford gates

• With 36 automorphisms and eight ancilla-supported Pauli measurements, 95 useful rotations generate the Clifford group on eleven of twelve logical qubits, while 288 logical measurements are available.The construction uses one logical qubit as a pivot for measurement-based synthesis and finds partially optimized implementations of all 411 Pauli rotations.Source: §4.3 · pp. 34–36
• 4.3 Implementing logical Clifford gates: Two or three measurements per rotation are required, because the final measurement of one rotation circuit can sometimes initialize the next.Using (65), three measurements per gate are required, but support-dependent reuse reduces this to two measurements per rotation for some sequences.Source: §4.3 · p. 36

• The gauge-fixed surgery scheme combines graph expansion, gauge fixing, and bridge systems to support low-overhead logical computation on QLDPC codes.Source: §5 · pp. 36–37
• Monolayer ancilla systems are affordable for some gross-code logical operators, but the analysis does not prove that one layer suffices for the gross code or other code families.Quantum expander codes are suggested as promising candidates for further study.Source: §5 · p. 37
• The paper leaves open whether its syndrome circuits are optimal in fault locations and whether efficient circuits can be designed for other measurement schemes.Source: §5 · p. 37
• Implementing logical computations may require many measurements and error-correction rounds, motivating investigation of single-shot error correction to reduce surgery time overhead.Source: §5 · p. 37

• Theorem 1 shows that the constructed code makes X-logical operator ¯XM a stabilizer and has exactly one fewer logical qubit than the original CSS code.Source: §A.1 · p. 37
• The proof establishes that the recursively defined X and Z checks commute, validating the merged code construction.Source: §A.1 · p. 37
• The construction uses equivalent representatives for gauge operators across odd layers to organize the recursive rank analysis.Source: §A.1 · p. 38
• The base case computes the check-matrix ranks and shows that adding one layer changes the logical-qubit count from k to k−1.Source: §A.1 · pp. 38–39
• Inductively, the added checks remain independent, while the remaining logical operators of the original code survive in the constructed code.Source: §A.1 · p. 39

• Theorem 20 establishes that the Y-system encodes k−1 logical qubits, corresponding to the original logical qubits retained from the input code.Source: §A.2 · pp. 39–40
• The Y-system proof removes bridge checks and qubits without changing the logical-qubit count, then analyzes the remaining symplectic check matrix.Source: §A.2 · p. 39
• For the base case, the non-CSS check is shown not to create a redundancy, enabling separate X- and Z-check rank bounds that imply k−1 logical qubits.Source: §A.2 · pp. 39–40
• Theorem 22 gives measurement fault distance at least d for bridge-connected joint measurements when both subsystems satisfy the stated Cheeger-constant conditions and R≥d rounds are used.Source: §A.2 · p. 40
• The fault-distance argument lower-bounds reduced supports through expansion and concludes that representative operators retain weight at least d.Source: §A.2 · pp. 40–41

Theorem 24

• For R ≥ d rounds, the bridged joint measurement system has measurement and logical fault distances at least d.This holds when both logical operators are minimal-weight representatives and their individual X-ancilla systems have measurement and logical fault distances at least d.Source: §A.3 · p. 41

Distance reduction

• The joint code’s fault distances reduce to proving that its X-distance is at least d, since the original code already supplies the required Z-distance bounds.The relevant distances are expressed as minima involving R, d_Z, and d_X; the proof establishes the remaining d_X bound.Source: §A.3 · p. 41

Block restrictions

• Any nontrivial restriction of a joint logical X operator to either block already has weight at least d.The restrictions commute with the checks of their respective merged codes and are therefore logical operators there.Source: §A.3 · p. 41

Bridge reduction

• If both block restrictions are trivial, stabilizers move the operator onto the bridge, where the repetition-code structure leaves only I or X(B).The nontrivial case has bridge weight |B| = min(|X̄1|, |X̄2|) ≥ d.Source: §A.3 · p. 41

Weight preservation

• Minimality of the block logical representatives prevents multiplying X(B) by stabilizers from reducing its weight.The proof bounds the resulting weight using triangle inequalities and the minimal-weight property, obtaining |X(B)S1| ≥ |B|.Source: §A.3 · p. 42

Syndrome circuits

• The numerical experiments use the original gross-code syndrome circuit and a two-cycle merged syndrome circuit for an X-ancilla system.The R, G, and B labels denote single CNOT layers determined by Tanner-graph edge coloring and corrected by the procedure in Figure 14.Source: Figure 13 · p. 43, I II III IV V VI VII VIII I II VIII · p. 43

Circuit correction

• Edge coloring produces an incorrect syndrome circuit in mixed-check regions because gauge-stabilizer support propagates onto check qubits.The correction identifies the responsible CNOT and moves it to the front of the circuit.Source: Figure 14 · p. 44, I II III IV V VI VII VIII I II VIII · p. 44