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Fault-tolerant logical measurements via homological measurement

Benjamin Ide, Manoj G. Gowda, Priya J. Nadkarni, Guillaume Dauphinais

arXiv:2410.02753v2quant-ph

TL;DR

The paper addresses the need for low-overhead, fault-tolerant measurement of logical Pauli operators in CSS and qLDPC codes. It develops homological measurement and an edge-expanded protocol based on mapping cones, then benchmarks the approach in a GKP-photonic architecture. The protocol supports arbitrary logical Pauli measurements with linear ancillary-qubit scaling, preserved distance, and logical error rates on par with methods using more ancillas.

  • Problem

    Fault-tolerant measurement of high-weight logical Pauli operators can require excessive resources, especially for general qLDPC codes and photonic architectures.

    Source: §I · p. 1, §III · p. 4

  • Method

    The paper uses chain-complex homological measurement and an edge-expanded mapping-cone construction to extend CSS codes while measuring target logical Pauli operators.

    Source: §I · pp. 1–2, §III · p. 4

  • Results

    The protocol measures arbitrary logical Pauli operators with ancillary-qubit overhead linear in operator weight while preserving code distance and achieving photonic logical error rates on par with higher-ancilla methods.

    Source: §I · p. 1, A. Mathematical properties of cone codes · p. 5, §V · pp. 15–16

  • Takeaways & Limitations

    Homological measurement provides a general language that includes lattice-surgery variants and enables lower-ancilla fault-tolerant measurements for CSS qLDPC codes.

    Source: §I · p. 2, §V · pp. 15–16

Abstract

from arXiv · show

We introduce homological measurement, a framework for measuring the logical Pauli operators encoded in CSS stabilizer codes. The framework is based on the algebraic description of such codes as chain complexes. Protocols such as lattice surgery some of its recent generalizations are shown to be special cases of homological measurement. Using this framework, we develop a specific protocol called edge expanded homological measurement for fault-tolerant measurement of arbitrary logical Pauli operators of general qLDPC codes, requiring a number of ancillary qubits growing only linearly with the weight of the logical operator measured, and guaranteed that the distance of the code is preserved. We further benchmark our protocol numerically in a photonic architecture based on GKP qubits, showing that the logical error rate of various codes are on par with other methods requiring more ancilla qubits.

I. INTRODUCTION

The paper motivates fault-tolerant logical measurements for qLDPC codes and introduces homological measurement as a chain-complex framework. It also defines the graph and hypergraph structures used to construct and analyze the protocol.

I. INTRODUCTION

  • qLDPC codes can reduce overhead relative to local topological codes, but photonic architectures still need low-space-overhead fault-tolerant logical measurement protocols.Photonic connectivity and photon-loss noise make these architectures compatible with qLDPC codes and measurement-based computation.Source: §I · p. 1
  • Existing generalized and homomorphic measurement methods impose substantial overhead, while CSS code surgery lacks guaranteed fault tolerance and is restricted to CSS logical Paulis.The cited refinement reduces overhead only for logical operators with suitable edge-expansion structure.Source: §I · p. 1
  • The proposed framework measures arbitrary multiqubit logical Pauli operators fault-tolerantly using ancillary-qubit overhead linear in the operator weight.The protocol extends the code while placing the measured logical operator in the stabilizer group.Source: §I · p. 1
  • Homological measurement represents CSS codes with chain complexes and uses homological algebra to describe logical operators and operations such as mapping cones.Lattice surgery and generalized lattice surgery arise as special cases within this framework.Source: §I · pp. 1–2

A. Graph theory

  • A hypergraph consists of vertices and edges that are arbitrary vertex subsets, and its incidence matrix represents edge–vertex membership.The paper generalizes graph-based Cheeger analysis to these incidence-matrix representations.Source: A. Graph theory · p. 2

B. Homological algebra

This section introduces finite chain complexes, their homology and cohomology, and the mapping-cone construction used to analyze logical measurements. Exact sequences provide the algebraic tools for tracking the resulting logical operators.

B. Homological algebra

  • A finite chain complex is a sequence of vector spaces and boundary maps whose consecutive composition is zero.Homology is formed as ker ∂i / im ∂i+1, while cohomology is defined from the dual cochain complex.Source: B. Homological algebra · pp. 2–3
  • Exact sequences characterize the relationship between images and kernels, with short and long exact sequences organizing these relationships across complexes.The rank-nullity theorem is generalized to exact sequences of finite-dimensional vector spaces.Source: B. Homological algebra · p. 3
  • Given a chain map f between complexes A and C, the mapping cone combines C_i with A_i−1 to form a new chain complex.The paper works over F2, so minus signs in the mapping-cone maps are omitted.Source: B. Homological algebra · p. 3
  • The mapping cone yields a short exact sequence and an associated long exact sequence on homology.These sequences allow the logical operators remaining after measurement to be studied algebraically.Source: B. Homological algebra · p. 3

C. Codes as chain complexes

CSS codes are equivalent to three-term chain complexes, allowing their stabilizers and logical operators to be expressed through boundary maps and homology. This equivalence enables new CSS codes to be constructed with mapping cones.

C. Codes as chain complexes

  • A classical linear code can be represented as a chain complex with one boundary map, and conversely any one-map chain complex defines a classical code.The parity-check matrix specifies the code parameters [n, k, d].Source: C. Codes as chain complexes · p. 3
  • For a CSS code, n counts physical qubits, k logical qubits, and d=min{dX,dZ}, where dX and dZ are minimum nontrivial logical-operator weights.The X- and Z-stabilizers are generated by the rows of HX and HZ.Source: C. Codes as chain complexes · p. 3
  • Two consecutive boundary maps define CSS check matrices, with X-logical operators represented by first homology and Z-logical operators by first cohomology.The chosen chain/cochain orientation is selected because it is natural for measuring X-logical operators.Source: C. Codes as chain complexes · p. 3
  • Because CSS codes and three-term chain complexes are equivalent, mapping cones can be used to construct new CSS codes.This supplies the algebraic basis for the measurement constructions developed later.Source: C. Codes as chain complexes · p. 3

III. HOMOLOGICAL MEASUREMENT

The paper frames logical measurement as a fault-tolerant code-extension problem: an ancillary complex and chain map place the target logical operator into the cone-code stabilizer group. Edge expanded homological measurement improves resource use by engineering edge expansion.

III. HOMOLOGICAL MEASUREMENT

  • The protocol targets fault-tolerant logical measurement with minimal ancillary-qubit requirements while preserving the remaining logical information.The extended code should maintain distance and keep stabilizer weights reasonably low.Source: §III · p. 4
  • Edge expanded homological measurement improves on generalized lattice surgery by directly improving edge expansion rather than relying on expansion already present in the Tanner graph.The construction applies to arbitrary X- or Z-logical operators and extends to mixed logical X-, Y-, and Z-Pauli operators.Source: §III · p. 4
  • Given a CSS code and target logical operator, the construction chooses an ancillary code and chain map so the operator becomes a stabilizer of the mapping-cone code.The measurement result must remain inferable despite errors, while other logical operators remain unmeasured.Source: §III · p. 4

A. Mathematical properties of cone codes

Cone-code homological measurement characterizes which original logical operators can be measured and proves conditions under which distance and logical information are preserved. The framework also recovers lattice surgery and generalized lattice-surgery constructions as special cases, while exposing their overhead and stabilizer-weight trade-offs.

Cone-code structure

  • The mapping-cone construction produces a CSS cone code whose new stabilizers can measure selected logical operators, while ker ∂1 characterizes whether additional logical operators are measured.The measured X operators form a space L, whereas X stabilizers form a space S; enforcing dim ker ∂1 = 1 prevents inadvertent extra measurements.Source: A. Mathematical properties of cone codes · pp. 4–5

Distance preservation

  • The cone-code framework guarantees that the dressed Z-distance is preserved, with edZ ≥ dZ, when ancilla-derived logical operators are treated as gauges.The proof reduces the dressed-distance analysis to a stabilizer code including a full basis of Z-gauges as stabilizers.Source: A. Mathematical properties of cone codes · p. 5

Known schemes

  • Lattice surgery is recovered by choosing an ancillary chain whose map contributes weight-two rows summing to the joint logical operator ¯X1 ¯X2, making it a stabilizer of the merged code.The resulting cone code is the merged system after surgery.Source: A. Mathematical properties of cone codes · pp. 5–6
  • Generalized lattice surgery arises from a repetition-code-based ancillary construction whose cone code measures a weight-w logical operator ¯X.The standard construction uses r = d to ensure distance preservation, while reduced-r variants use Tanner-graph edge expansion to bound admissible choices.Source: A. Mathematical properties of cone codes · p. 6

Limitations of prior schemes

  • The reduced-r generalized-surgery scheme can introduce new ∂0 rows that are not low weight, including weight-d rows when measuring a weight-d logical operator in a distance-d toric code.Thus, the construction does not always remain LDPC in its added stabilizers.Source: A. Mathematical properties of cone codes · p. 6

Framework scope

  • Homological measurement unifies lattice surgery, generalized lattice surgery, and related individual-logical schemes, although their joint-measurement procedures differ.The paper uses this unified cone-code viewpoint to develop a more optimized measurement scheme afterward.Source: A. Mathematical properties of cone codes · pp. 5–6

B. Measuring an X-logical operator

This section develops homological measurements for X-logical operators through chain-map constructions involving edge expansion, graph augmentation, cycle-basis selection, and cellulation. It examines how these choices affect distance preservation, stabilizer weights, ancilla overhead, connectivity, and parallel-measurement behavior.

Construction

  • The construction starts from HZ restricted to the measured support, expands hyperedges into ordinary edges, raises the Cheeger constant to 1, and selects sparse cycle-basis rows for ∂0.Algorithm 2 searches randomly for low-weight ∂0 choices, while cellulation can split large cycles when no sufficiently sparse cycle basis exists.Source: B. Measuring an X-logical operator · pp. 6–9

Distance preservation

  • The cone code preserves the original X-distance when the graph induced by ∂1 has Cheeger constant at least 1.The construction also retains the original logical operators while ensuring the measured operator becomes a stabilizer.Source: B. Measuring an X-logical operator · pp. 7, 10

Target selectivity

  • A positive Cheeger constant makes the graph connected, so dim ker ∂1 = 1 and only the desired logical operator is measured, including for joint measurements.Joint measurements therefore require no additional work after applying Algorithm 1.Source: B. Measuring an X-logical operator · p. 7

Sparsity and connectivity

  • For qLDPC input codes, ∂0 is not guaranteed to be sparse; improving its sparsity can begin by replacing hyperedges with collections of ordinary two-vertex edges.A decongestion-based cycle basis increases cone-code connectivity at most polylogarithmically in the measured logical operator’s weight relative to the input code; Algorithm 2 is suggested for practical use.Source: B. Measuring an X-logical operator · pp. 8–9

Parallel measurement

  • Independent measurements can be performed iteratively, but parallel X- and Z-type measurements require accounting for the Z operator’s support extending onto ancillas.After an X-type measurement, the remaining X-logical operators are unchanged.Source: B. Measuring an X-logical operator · p. 9

Resource example

  • For a weight-8 logical operator, edge-expanded homological measurement uses 10 ancillas without cellulation or 12 with cellulation, versus 24 and 120 for two alternative constructions.The edge-expanded construction has maximum stabilizer weight 5; the comparison assumes code distance d = 8.Source: B. Measuring an X-logical operator · p. 9

Construction

  • The construction expands hyperedges into weight-two edges while preserving the Cheeger constant as high as possible, then computes a cycle basis and cellulates oversized cycles.It applies Algorithms 1 and 2 to update incidence matrices and mappings before defining the mapping cone.Source: B. Measuring an X-logical operator · p. 10

Steane code

Quantum Hamming code

  • For the quantum Hamming code, omitting Algorithm 1 produces a [[19, 6, 2]] code, demonstrating that a low Cheeger constant can decrease distance.The logical operator is X3X4X5X12X14.Source: B. Measuring an X-logical operator · p. 11
  • Only two additional edges, hence two ancilla, are needed to maintain distance for the quantum Hamming-code measurement.Without Algorithm 1, the construction produces a [[19, 6, 2]] code with Cheeger constant 0.5 and weight-two logical operators.Source: B. Measuring an X-logical operator · p. 11

C. Fault tolerant protocol for homological measurements

The protocol turns the chain-complex construction into a fault-tolerant procedure for extending a CSS code with the measured logical operator. It measures new stabilizers repeatedly, decodes their outcomes, and safely removes the ancillas.

Protocol overview

  • The framework itself specifies the final code but not an implementation protocol, so the section gives steps from HX and HZ to eHX and eHZ.The procedure applies to homological measurements generally, not only edge expanded homological measurement.Source: C. Fault tolerant protocol for… · p. 11

Ancilla initialization and stabilizer measurement

  • Ancillas are initialized in |0⟩ for X measurements or |+⟩ for Z measurements before the new X stabilizers are measured and their random outcomes recorded.The construction ensures the relevant existing measurements commute with the new measurements.Source: C. Fault tolerant protocol for… · p. 11

Gauge handling

  • Z stabilizers are measured selectively, treating stabilizers outside rowspace(∂0) as fixed gauges to avoid potentially high-weight generators.Only the required stabilizers are measured when rowspace(∂0) is a proper subset of rowspace(G).Source: C. Fault tolerant protocol for… · p. 12

Fault tolerance

  • New-stabilizer outcomes are repeated over time and decoded because their products reveal the logical measurement while individual results can be random or error-prone.Repetition is typically tied to the initial code distance and also protects the other logical qubits.Source: C. Fault tolerant protocol for… · p. 12

Ancilla removal

  • The ancillas are finally measured in the Z basis to determine the resulting code space, and this readout must itself be fault tolerant.The ancilla measurements can contain errors.Source: C. Fault tolerant protocol for… · p. 12

D. Jointly measuring a mixture of X-, Y -, and Z-logical operators

The procedure extends separate X- and Z-logical measurements to arbitrary mixtures of X, Y, and Z logical operators. It merges stabilizers to resolve anticommutation or unwanted separate measurements, then removes undesired low-weight logical operators while preserving distance.

Mixed-Pauli decomposition

  • An arbitrary mixed logical Pauli is decomposed into X- and Z-logical factors, with an overall sign tracked to interpret the measurement correctly.Algorithm 1 is applied separately to the X and Z components.Source: D. Jointly measuring a mixture of X-,… · p. 12

Stabilizer merging

  • The initial matrices may anticommute or measure X and Z separately, so stabilizer pairs are merged to obtain the desired joint measurement.Anticommuting pairs arise from generators with a single overlapping qubit, while commuting cases merge X and Z stabilizers across the relevant rows.Source: D. Jointly measuring a mixture of X-,… · p. 12

Removing unwanted logical operators

  • The merged construction implements the logical operator but can introduce new low-weight logical operators, which are identified as cycles of a graph formed from merged stabilizers.The graph combines the graphs associated with the relevant boundary maps.Source: D. Jointly measuring a mixture of X-,… · p. 12

Sparsity control

Distance guarantee

  • The resulting code loses one logical dimension by construction, while its distance is preserved and may increase because the complementary distance can grow.The distance argument decomposes errors into X and Z parts and applies the theorem independently.Source: D. Jointly measuring a mixture of X-,… · p. 13

IV. EXAMPLES AND NUMERICAL SIMULATIONS

The section introduces numerical benchmarking of the fault-tolerant logical-measurement protocol in a measurement-based quantum computer using GKP qubits concatenated with quantum LDPC codes.

Numerical simulations

  • The numerical study benchmarks the fault-tolerant logical-operator measurement protocol in a measurement-based quantum computer.Source: §IV · p. 13

Photonic architecture

  • The simulated architecture uses GKP qubits concatenated in various quantum LDPC codes.Source: §IV · p. 13

Evaluation scope

  • The simulations evaluate code behavior after logical measurements have been performed on representative quantum LDPC codes.Source: §IV · p. 13

A. Examples

The examples study lifted-product and hypergraph-product qLDPC codes, chosen for scaling, constant encoding rate, and compatibility with the photonic architecture. They measure randomly selected X-logical operators whose weights equal the code distances and report post-measurement code parameters.

Code families

  • The examples use lifted-product and hypergraph-product qLDPC codes because their distances scale with n, their encoding rates are constant, and they perform well in the target photonic architecture.The homological-measurement procedure itself applies to any code.Source: A. Examples · p. 13

Measured operators

  • Each example measures a randomly chosen X-logical operator whose weight equals the distance of its original code.Source: A. Examples · p. 13

Lifted-product construction

  • Lifted-product codes are constructed as lifted versions of hypergraph-product codes using polynomial quotient rings and circulant matrix lifts.The simulations use matrices whose entries are monomials in the quotient ring.Source: A. Examples · p. 13

Hypergraph-product construction

  • The hypergraph-product examples use identical classical parity-check matrices with column weight four and row weight three, obtained using the MacKay–Neal method.Source: A. Examples · pp. 13–14

Reported parameters

  • The examples report measured logical operators in Table II, original-code parameters in Table III, and post-measurement parameters and weights in Table IV.Source: A. Examples · p. 14

B. Numerical simulations

The simulations benchmark homological measurement in a photonic GKP architecture using foliated codes, Monte Carlo logical block error rates, and BP-OSD decoding. Across lifted product and hypergraph product codes, the protocol retains performance comparable to existing methods while using fewer ancillas in the reported comparisons.

Simulation architecture

  • The simulations use foliated quantum codes in an optical GKP architecture with beamsplitter-based cluster-state generation and homodyne measurements.The code is foliated with a number of layers equal to its distance before constructing the cluster state.Source: B. Numerical simulations · p. 14

Decoding

  • The decoder uses belief propagation with a flooding schedule and OSD fallback, making the procedure applicable across quantum LDPC codes.OSD uses combination sweep with search depth λ = 60 when BP does not converge.Source: B. Numerical simulations · p. 15

Performance metric

  • Performance is quantified by logical block error rate estimated from Monte Carlo trials before and after homological measurement.A trial counts as failed when any logical qubit remains erroneous after decoding, with Agresti-Coull uncertainty estimates at a 95% confidence level.Source: B. Numerical simulations · p. 15

Lifted product codes

  • For lifted product codes LP1 and LP2, the protocol performs similarly to with fewer additional ancilla qubits, while block error rates remain similar to the original codes.Figure 4 reports the before-and-after logical block error rates for these codes.Source: B. Numerical simulations · p. 15

Hypergraph product codes

  • For hypergraph product codes HGP1 and HGP2, the protocol performs equivalently to and outperforms generalized lattice surgery.The reported explanation is that generalized lattice surgery repeats a potentially poor structure d times and uses a significantly larger ancilla code.Source: B. Numerical simulations · p. 15

V. DISCUSSION

The discussion presents edge expanded homological measurement as a general CSS-code framework that reduces ancilla requirements while maintaining strong simulated logical performance. It also identifies unresolved trade-offs involving optimization, time overhead, qLDPC preservation, and parallelization.

Framework and protocol

  • The framework yields fault-tolerant logical Pauli measurement with fewer ancilla qubits than generalized lattice surgery, its refinement, and homomorphic measurements.The protocol is described as applicable to any CSS code and is called edge expanded homological measurement.Source: §V · pp. 15–16

Comparison with related work

  • The ancilla comparison is presented as a trade-off against related work [68], which generally uses more ancillas in exchange for a provably qLDPC construction.The discussion examines a special single-layer case and describes a Cheeger-constant-based optimization within the homological framework.Source: §V · p. 16

Numerical evidence

  • The simulations show robust logical performance despite the slight connectivity increase of a limited number of stabilizer generators.The discussion connects this result to the architecture’s sensitivity to stabilizer measurement weight and connectivity.Source: §V · p. 16, Figure 5 · p. 17

Open optimization

  • The reported analysis does not optimize logical-operator representatives or decoding strategies for the structures introduced by the protocol.Optimization for specific logical-operator structures is left for future work.Source: §V · p. 17

Time overhead

  • The protocol minimizes ancilla count but does not reduce time overhead, requiring roughly d rounds of new-stabilizer measurements for reliable inference.The discussion suggests that single-shot error correction or longer-chain-complex constructions could motivate constant-round schemes.Source: §V · p. 17

qLDPC preservation

  • The construction does not guarantee that an input qLDPC code remains qLDPC, although the specific near-term constructions had low practical connectivity.Worst-case connectivity is stated to increase polylogarithmically with distance, while possible modifications and weight-reduction techniques are left for future work.Source: §V · p. 17

Parallelization

  • Parallel measurement of commuting logical Pauli operators is straightforward, but increasing the number measured generally raises connectivity and limits practical parallelization.The achievable amount depends on the specific codes and logical operators.Source: §V · p. 17

Appendix A: The mapping cylinder

The appendix examines mapping-cylinder constructions as an alternative to the mapping cone and explains why they do not yield a useful logical-measurement scheme. Although Z-distance can be preserved, the construction fails to realize the desired logical measurement under the available choices.

Appendix A: The mapping cylinder

  • The mapping-cylinder construction was investigated but did not produce a useful scheme for implementing logical measurements.The appendix presents the construction and its failure as a possible direction for future work.Source: Appendix A · p. 18
  • The mapping cylinder of a chain map f : A → C is defined as a chain complex combining C_i, A_{i−1}, and A_i, and it can also be viewed as a mapping cone.The appendix states that cyl(f) = cone(g) for a different chain map g.Source: Appendix A · p. 18
  • The cylinder code is formed from a CSS code and a chain map f : A → C, with its stabilizer matrices obtained from the corresponding chain-complex construction.The appendix introduces the cylinder code through the CSS chain and the chain map before discussing its stabilizers.Source: Appendix A · p. 18
  • Z-distance is maintained, but the two possible choices for ∂2 do not yield a useful measurement: one makes the added operator as heavy as the target logical, while the other excludes it from the relevant row space.The X-distance is also maintained under the stated assumptions, independently of the graph’s Cheeger constant.Source: Appendix A · p. 18
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