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Dynamically Generated Logical Qubits
Matthew B. Hastings, Jeongwan Haah
TL;DR
The paper addresses how a code with no subsystem-code logical qubits can nevertheless support logical information and fault-tolerant memory. It develops dynamically generated logical qubits through two-qubit Pauli measurement patterns, establishing toric-code-like structure while identifying boundary constraints and time-boundary requirements.
Problem
The paper asks how fault-tolerant error correction and logical information can arise when the code, viewed as a subsystem code, has no logical qubits.
Method
The authors use periodic sequences of two-qubit Pauli measurements to define instantaneous stabilizer groups, logical operators, and toric-code-like dynamics.
Results
The measurement sequence generates a four-dimensional stabilized subspace with four outer logical operators, while each instantaneous stabilizer group is equivalent to a toric code up to a bounded-depth circuit.
Takeaways & Limitations
The construction provides a model of a fault-tolerant quantum memory with dynamically generated logical qubits and toric-code-like behavior.
Takeaways & Limitations
Fault-tolerance proofs for noisy measurements initially assume indefinite code dynamics; operational use therefore requires logical initialization and destructive measurements, while pairwise boundary checks face a topological obstruction.
Abstract
from arXiv · showhide
We present a quantum error correcting code with dynamically generated logical qubits. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, our measurement patterns generate logical qubits, allowing the code to act as a fault-tolerant quantum memory. Our particular code gives a model very similar to the two-dimensional toric code, but each measurement is a two-qubit Pauli measurement.
1 The Honeycomb Code
The honeycomb code uses a time-ordered sequence of two-qubit Pauli measurements to create a dynamic two-qubit logical subspace, despite having no subsystem-code logical operators. Its instantaneous stabilizer dynamics realize toric-code structure and transform logical operators over time.
- 1.1 The Code: The honeycomb code defines logical qubits through a time-ordered sequence of two-qubit Pauli checks, rather than a fixed encoded subspace.At any moment, the instantaneous stabilizer group identifies a four-dimensional subspace containing two logical qubits.
- 1.2 Properties as Subsystem Code: The subsystem-code formulation has no logical operators because the gauge group and stabilizer group account for all physical qubits.For a torus with n_p plaquettes, the gauge group has dimension 3n_p − 1, the stabilizer group has dimension n_p + 1, and there are n_p − 1 gauge qubits.
- 1.3 Instantaneous Stabilizer Groups: After three measurement rounds, the instantaneous stabilizer group is periodic and stabilizes a four-dimensional subspace, but the measurement ordering is essential.An alternative schedule measuring x, y, z by round would place long loop operators in the instantaneous stabilizer group and destroy the logical subspace.
- 1.4 Logical operators and embedded toric code: A depth-1 disentangling circuit maps the instantaneous stabilizer state to a toric code on a hexagonal superlattice, enabling toric-code string operators to represent honeycomb-code logical operators.The constructed strings act as magnetic or electric operators, depending on whether they transport superplaquette excitations of type m or the corresponding electric excitation.
- 1.4 Logical operators and embedded toric code: Measurement dynamics map magnetic logical operators to electric ones and vice versa, while a three-round period multiplies each outer logical operator by a parallel inner logical operator.Because the measurement sequence has period 3 but the outer-operator dynamics have period 6, inner operators carry the residual transformation; their commutation relation identifies them as fermion-transporting operators.
2 Ladder Code
The ladder code uses a four-round sequence of two-qubit measurements to generate logical qubits dynamically. Its syndrome measurements detect single-qubit and check errors, while decoding can correct sufficiently sparse errors under stated conditions.
- Code definition: The ladder has one qubit per vertex, with vertical ZZ checks and alternating horizontal XX and YY checks on a periodic even-rung geometry.
- Measurement dynamics: The repeating schedule measures ZZ, XX, ZZ, and YY checks in rounds r = 0, 1, 2, 3 mod 4, respectively.The four-round pattern avoids measuring an inner logical operator that would arise under a three-round schedule.
- Instantaneous stabilizer group: For r ≥4, the instantaneous stabilizer group is generated by square plaquette stabilizers and the most recently measured checks.
- Logical operators: The inner logical operator is a product of checks along one ladder leg, while outer logical operators are products of XX or YY checks on a rung, depending on the round.
- Fault tolerance: Single-qubit Pauli errors flip at least one plaquette syndrome, and the ladder therefore detects them with effective distance 2.The instantaneous stabilizer code itself has code distance 2.
- Fault tolerance: At low error rates, minimum-weight matching can correct sufficiently small chains of check errors, including when syndrome bits form a 1 + 1-dimensional spacetime array under measurement errors.For persistent errors near readout, majority decoding across L positions can recover the outer logical operator.
3 Fault-tolerance of the honeycomb code on torus
The honeycomb code’s syndrome bits form a structured spacetime decoding graph, enabling minimum-weight matching to correct faults below a positive threshold and support logical readout with finite-time boundaries.
- Error model: 18 independent elementary faults occur per unit spacetime volume in the simplified error model.The model restricts Pauli faults according to the checks measured immediately before and after each error.
- Syndrome geometry: Each single-qubit fault produces a pair of syndrome-bit changes whose allowed separations generate the decoding graph.Rotational and time-translation symmetry restrict possible pairs to integer combinations of s1 + s2, s2 + s3, and s3 + s1.
- Syndrome geometry: The decoding graph consists of two shear-transformed simple-cubic lattices, reflecting the even sublattice covered by the syndrome-change generators.The cubic lattice’s (111) direction corresponds to time.
- Fault tolerance: Minimum-weight matching identifies faults up to plaquettes, and a Peierls argument establishes a positive threshold below which residual error loops remain small.All Pauli and measurement-outcome faults can be represented as 1-chains, while syndrome changes form 0-chains on the decoding graph.
- Fault tolerance: Null-homologous decoding cycles decompose into square cycles that are inconsequential to later logical operators, yielding positive fault tolerance.The square cycles correspond to the inconsequential error configurations illustrated in Figure 6.
- Boundaries and logical readout: The fault-tolerance proof assumes indefinitely continuing code dynamics, so meaningful time boundaries are introduced for logical initialization and destructive measurement.Logical X- and Z-basis measurements use a strategy based on the honeycomb code’s equivalence to the toric-code state.
4 Boundary Conditions
The honeycomb code faces a topological obstruction when introducing boundaries while retaining cyclic bulk measurements using only pairwise checks. Boundary constructions therefore require careful measurement schedules or more complex operations.
- Boundary construction: The proposed planar setting uses an annulus with armchair boundary conditions and added edges and square plaquettes to maintain trivalence.The added edges are type 0, the added plaquettes are type 2, and their checks remain two-qubit Pauli products.
- Boundary construction: The boundary ISG adds square-plaquette operators and, for r = 2 mod 3, checks on added type 2 edges.These generators produce smooth boundaries for r = 1 mod 3 and rough boundaries for r = 0, 2 mod 3.
- Boundary dynamics: Measuring type r + 1 checks is compatible with the boundary for r = 0, 1 mod 3 but problematic for r = 2 mod 3.The problematic transition measures the inner logical operator, expressed as the product of checks along the bottom boundary.
- Topological obstruction: A dimer representation tracks the instantaneous stabilizer group, with each generator represented by an unordered pair and measurements rewiring paired sites.The number of dimers crossing a vertical cut is invariant modulo 2.
- Topological obstruction: Using a naive boundary while applying the 0, 1, 2 sequence causes a nontrivial homology change and reveals the inner logical operator at the opposite boundary.The obstruction follows because the boundary path sum must acquire the same nontrivial homology representative at the other edge.
- Workaround: The obstruction can be avoided with nonpairwise boundary checks, implemented using ancillas, together with boundary shrinking and regrowth.This permits cyclic bulk transitions while changing boundary type during the measurement sequence.
A A Toy Model of Dynamical Quantum Memory: Not an Error Correcting Code
A toy model combining random measurements with fast scrambling can preserve entanglement for long times, but it does not provide useful error correction against external noise. Its measurement outcomes rarely reveal that noise occurred.
- Behavior without external noise: Fast-scrambling monitored models can exhibit a measurement-rate transition between rapid purification and long-lived mixed states.With less frequent measurements, a mixed initial state can remain mixed for very long times, and reference entanglement can persist.
- Toy model: The toy model alternates single-qubit Z measurements with random global Clifford operations, equivalently measuring uniformly random nonidentity Pauli products.The number of stabilizers increases only when a measurement commutes with the existing stabilizer group.
- Behavior without external noise: The time for the system to become pure, or for system-reference entanglement to disappear, is exponential in N.The entanglement statement assumes an initially maximally entangled pure state and no external noise.
- Failure under external noise: External random Clifford noise is detected with only exponentially small probability because measured operators almost never belong to the stabilizer group.Consequently, noisy and noiseless measurement outcomes have the same distribution except with exponentially small probability.