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Low-overhead fault-tolerant quantum computation by gauging logical operators

Dominic J. Williamson, Theodore J. Yoder

arXiv:2410.02213v2quant-phcond-mat.str-el

TL;DR

Existing fault-tolerant logical-measurement schemes on sparse quantum memories do not always achieve low qubit overhead. The procedure treats a logical operator as a physical symmetry and gauges it through local measurements, using code deformation and repeated syndrome measurements. The two-gross code measurement adds 18 X checks, 13 Z checks, and 34 qubits, totaling 65; the deformation has minimal weight-7 checks and degree-7 qubits without higher connectivity.

Abstract

from arXiv · show

Quantum computation must be performed in a fault-tolerant manner to be useful in practice. Recent progress has established quantum error-correcting codes with sparse connectivity requirements and constant qubit overhead suitable for quantum memory. However, existing schemes that include fault-tolerant logical measurement on such quantum memories do not always achieve low qubit overhead. Here we present a low-overhead method to implement fault-tolerant logical measurement on a quantum error-correcting code by treating the logical operator as a physical symmetry and gauging it so that it is enforced by a product of local symmetries. The gauging measurement procedure introduces a high degree of flexibility that can be exploited to achieve a qubit overhead that is linear in the weight of the operator being measured up to a polylogarithmic factor. This flexibility also allows the procedure to be adapted to arbitrary quantum codes. Our results provide a more efficient approach to performing fault-tolerant quantum computation, making it more tractable for near-term implementation.

COMPETING INTERESTS

A patent application contains technical aspects from this paper.

DATA AVAILABILITY

The work generated no data.

BB CODE EXAMPLES

The BB-code construction uses polynomial matrices and partitions physical qubits into left and right sets with corresponding checks.

  • The construction defines x and y from cyclic permutation matrices, satisfying x^ℓ = y^m = I_ℓm and x^⊤x = y^⊤y = I_ℓm.Source: BB CODE EXAMPLES · p. 13
  • BB codes are CSS codes with n = 2ℓm physical qubits divided into ℓm left and ℓm right qubits.Source: BB CODE EXAMPLES · p. 13
  • Each X check acts on left and right qubits through polynomial matrices A and B, while each Z check uses B^⊤ and A^⊤.Source: BB CODE EXAMPLES · p. 13

Gross code

The gross-code construction uses gauging-measurement flexibility to measure a logical operator while preserving distance and limiting added connectivity.

Gross code

  • The gross code is a J144, 12, 12K bivariate-bicycle code, named because a gross is a dozen dozens or 144.Source: Gross code · p. 13
  • The construction measures Xα with fewer additional qubits and checks than existing constructions without decongestion or cellulation.Source: Gross code · p. 14
  • The design avoids new checks or qubits with Tanner-graph degree above 6, although 12 target qubits and 18 adjacent Z checks necessarily reach degree 7.Source: Gross code · p. 14
  • The graph starts with 18 edges from shared-check adjacency, then gains random edges selected to improve expansion and distance.Source: Gross code · p. 14
  • Four additional graph edges suffice to produce a deformed gross code with distance 12, matching the original code distance.Source: Gross code · p. 14
  • The 22-edge graph has 11 basis cycles, but four are redundant, leaving 7 independent flux checks in B.Source: Gross code · p. 14
  • 41 additional checks and qubits implement the gross-code deformation: 12 X checks, 7 Z checks, and 22 auxiliary qubits.Source: Gross code · p. 14

Two-gross code

  • The gauging-measurement framework is further demonstrated using the J288, 12, 18K BB code from Ref., referred to as the two-gross code.Source: Gross code · p. 14

Two-gross code

The two-gross code measurement constructs a suitable auxiliary graph for a weight-18 logical operator and preserves low check weights and qubit degrees in the deformed code.

Two-gross code

  • The two-gross code uses weight-18 logical operators Xα = X(αf, 0) for α ∈ M.Source: Two-gross code · p. 15
  • The auxiliary graph connects vertices corresponding to monomial terms in f, adding 27 edges to ensure a sparse matching matrix.Source: Two-gross code · p. 15
  • Seven additional edges are added to ensure the deformed code has distance 18, producing a multigraph because two edges connect the same vertices.Source: Two-gross code · p. 15
  • The construction completes the graph with a basis of 13 cycles.Source: Two-gross code · p. 15
  • The deformation adds 18 X checks, 13 Z checks, and 34 qubits, totaling 65, while requiring no higher connectivity than the original construction.Source: Two-gross code · p. 16, Table II · p. 16

RECOVERING EXISTING PROTOCOLS FROM THE GAUGING MEASUREMENT FRAMEWORK

The gauging framework recovers lattice surgery, Shor-style measurement, and other logical-measurement schemes as special choices of graphs or hypergraphs. It also extends to simultaneous operator measurement and code initialization or stabilizer measurement.

RECOVERING EXISTING PROTOCOLS FROM THE GAUGING MEASUREMENT FRAMEWORK

GENERALIZATIONS OF THE GAUGING MEASUREMENT PROCEDURE

The gauging measurement procedure supports flexible graph, hypergraph, parallel, and circuit-based generalizations. These extensions reduce auxiliary resources or broaden the measurable operators, but some alternatives sacrifice scalability or code distance.

GENERALIZATIONS OF THE GAUGING MEASUREMENT PROCEDURE

  • A tensor-product representation lets each auxiliary vertex connect to several logical-support qubits, reducing a weight-d logical measurement from d to d/c auxiliary vertices when the graph has sufficient expansion.Source: GENERALIZATIONS OF THE GAUGING… · p. 17
  • The procedure extends to non-Pauli operators, qudits, and nonabelian groups, although nonabelian gauging does not fix a definite global charge through local charge measurements.Source: GENERALIZATIONS OF THE GAUGING… · p. 17
  • Multiple logical operators can be measured in parallel when no pair applies different nontrivial Paulis to a shared qubit, with constant support overlap required to maintain LDPC structure.Source: GENERALIZATIONS OF THE GAUGING… · pp. 17–18
  • Equivalent logical measurements can trade space for time by using 2^m − 1 parallel measurements over d^m rounds and taking a majority vote.Source: GENERALIZATIONS OF THE GAUGING… · p. 18
  • Replacing the auxiliary graph with a hypergraph introduces one qubit per hyperedge and measures vertex-hyperedge X checks for simultaneous abelian operator measurement.Source: GENERALIZATIONS OF THE GAUGING… · p. 18
  • Inferring high-weight plaquette checks only from initialization and readout preserves fault-distance scaling but creates large detector cells and is not expected to have a threshold without further modifications.Source: GENERALIZATIONS OF THE GAUGING… · p. 18
  • A circuit implementation uses entangling gates, vertex X measurements, and edge Z measurements; discarding decoupled vertices can reduce distance by a constant factor, avoidable with an extra dummy vertex.Source: GENERALIZATIONS OF THE GAUGING… · p. 18

SPACETIME CODE AND SPACETIME FAULT-DISTANCE

The spacetime construction models gauging measurement through repeated syndrome rounds around code deformation, using local detectors and stabilizers to characterize fault syndromes. It establishes fault-distance conditions for measurement and initialization based on deformation timing, graph expansion, and the number of repeated-measurement rounds.

Spacetime code

  • The fault-tolerant procedure uses at least d syndrome rounds before deformation, during the deformed code, and after returning to the original code.The sequence starts at t_0, deforms at t_i, ungauges at t_o, and includes repeated measurements in each code phase.Source: SPACETIME CODE AND SPACETIME… · p. 18
  • Local detectors are generated by repeated check measurements plus boundary detectors involving edge-qubit initialization at t_i and readout at t_o.The detector decomposition assumes no local relations in the original code; codes with local relations require adding space-only detectors.Source: SPACETIME CODE AND SPACETIME… · p. 19

Spacetime syndromes

  • During the deformed interval, vertex and edge Pauli faults create distinct detector syndromes, while measurement faults propagate syndromes across time and A_v syndromes can condense at deformation boundaries.Z_e faults move A_v syndromes along graph edges, X_e faults produce B_p and possibly clustered s̃_j syndromes, and deformation maps B_p or s̃_j syndromes into vertex errors up to spacetime stabilizers.Source: SPACETIME CODE AND SPACETIME… · p. 20

Spacetime stabilizers

  • The analysis defines spacetime logical faults as detector-free fault collections and spacetime stabilizers as detector-free collections that do not alter the gauging result.Local spacetime stabilizers are generated by code checks, paired Pauli faults with compensating measurement faults, and initialization or readout equivalents.Source: SPACETIME CODE AND SPACETIME… · pp. 20–22

Spacetime fault-distance

  • The measurement and initialization fault-distance is (t_o − t_i), because repeated faults on an A_v check across all intermediate rounds change the inferred logical measurement.The lower bound is saturated by this repeated A_v-measurement fault, while corresponding B_p and s̃_j fault collections are spacetime stabilizers.Source: SPACETIME CODE AND SPACETIME… · p. 22
  • The procedure includes d repeated stabilizer-measurement rounds before and after code deformation when perfect endpoint measurements are not assumed.These rounds prevent additional logical measurement faults from extending between the deformation endpoints and the whole procedure’s boundaries.Source: SPACETIME CODE AND SPACETIME… · p. 22
  • Any spacetime logical fault is equivalent, up to spacetime stabilizers, to the product of a space logical fault and a time logical fault.The proof cleans the space component to an initial timestep and converts measurement-fault strings into timelike logical faults, leaving a residual spacelike logical fault.Source: SPACETIME CODE AND SPACETIME… · p. 22
  • The spacetime fault-distance of the fault-tolerant gauging measurement procedure is d when h(G) ≥1 and (to −ti) ≥d.Logical faults that are not equivalent to spacelike faults span the intermediate time steps, while equivalent faults inherit the space-distance lower bound d.Source: SPACETIME CODE AND SPACETIME… · pp. 22–23
  • The distance guarantee also applies to high-weight or infrequently measured plaquette checks, and even to detectors inferred only from initialization and final readout.The inference-only variant avoids measuring high-weight operators but is likely not scalable against uncorrelated random Pauli and measurement noise on all fault sites.Source: SPACETIME CODE AND SPACETIME… · p. 23
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