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Mitigating Errors in Local Fermionic Encodings

Johannes Bausch, Toby Cubitt, Charles Derby, Joel Klassen

arXiv:2003.07125v2quant-ph

TL;DR

Low-weight fermionic encodings make local lattice operators practical but conflict with high-distance error correction. The paper analyzes local qubit noise in the VC and LW encodings and finds that many single-qubit errors are detectable, while most undetectable ones become local fermionic phase noise. Corner Majorana exceptions require targeted mitigation, so useful suppression can be achieved while accepting natural fermionic noise.

  • Problem

    Fermionic encodings face a trade-off between low-weight local-operator representations and high-distance error-detecting code spaces.

    Source: §1 · p. 2

  • Method

    The paper analyzes iid local qubit noise in the VC and LW fermionic encodings and characterizes the fermionic operators associated with undetectable errors.

    Source: §1 · p. 3

  • Results

    Many weight-1 errors are detectable, while almost all remaining undetectable errors map to local fermionic phase noise, with corner-qubit exceptions producing single Majorana errors.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Low-weight fermionic encodings can provide useful error mitigation when simulated natural fermionic noise is acceptable.

    Source: §1 · pp. 2–3

    The analysis assumes iid local qubit noise and leaves corner Majorana cases that can violate fermionic parity superselection as exceptions requiring mitigation.

    Source: §1 · p. 3

Abstract

from arXiv · show

Quantum simulations of fermionic many-body systems crucially rely on mappings from indistinguishable fermions to distinguishable qubits. The non-local structure of fermionic Fock space necessitates encodings that either map local fermionic operators to non-local qubit operators, or encode the fermionic representation in a long-range entangled code space. In this latter case, there is an unavoidable trade-off between two desirable properties of the encoding: low weight representations of local fermionic operators, and a high distance code space. Here it is argued that despite this fundamental limitation, fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular when undetectable errors correspond to "natural" fermionic noise. We illustrate this point explicitly for two fermionic encodings: the Verstraete-Cirac encoding, and an encoding appearing in concurrent work by Derby and Klassen. In these encodings many, but not all, single-qubit errors can be detected. However we show that the remaining undetectable single-qubit errors map to local, low-weight fermionic phase noise. We argue that such noise is natural for fermionic lattice models. This suggests that even when employing low-weight fermionic encodings, error rates can be suppressed in a similar fashion to high distance codes, provided one is willing to accept simulated natural fermionic noise in their simulated fermionic system.

1. Introduction and Summary of Results

The paper studies whether low-weight fermionic encodings can mitigate local qubit noise despite the tension between low-weight operators and strong error-detecting codes. For the VC and LW encodings, many single-qubit errors are detectable, while most remaining errors become local fermionic phase noise, with corner Majorana exceptions requiring mitigation.

Motivation

  • Low-weight fermionic operators conflict fundamentally with high-distance error correction because low-weight logical operators can occur as physical errors.The paper therefore evaluates error mitigation rather than requiring every error to be detectable or correctable.Source: §1 · p. 2

Scope and motivation

  • The VC and LW encodings are examined as low-weight mappings for local fermionic lattice Hamiltonians, including terms relevant to the Fermi-Hubbard model.Their low-weight representations are useful for near-term simulation, where higher-weight terms can increase circuit depth.Source: §1 · pp. 2–3

Noise model

  • The noise model assumes iid local qubit noise, while the reported salient behavior depends on locality rather than the specific noise operators.The model is described as simplistic but a surprisingly good match to noise observed in current hardware.Source: §1 · p. 3

Main results

  • Many weight-1 errors are detectable in both the VC and LW encodings because they do not commute with stabilizers.This provides error-detecting behavior without abandoning low-weight fermionic operator representations.Source: §1 · pp. 2–3
  • Almost all remaining undetectable weight-1 errors map to local fermionic phase noise, except corner-qubit cases that can produce single Majorana errors in both encodings.The paper also explores mitigation of weight-2 errors and strategies for the exceptional corner cases.Source: §1 · p. 3

Mitigating edge cases

  • Single Majorana corner errors can produce unphysical fermionic states that violate parity superselection, motivating lattice modifications or global fermionic-parity stabilizer checks.In the LW encoding, modifying the corners can convert these cases into weight-2 qubit errors; parity checks may be measured only at the end.Source: §1 · pp. 3–4

Significance

  • These error-mitigation properties are especially relevant to near-term circuits, where error correction is unavailable and faithful execution requires far fewer than one error during the circuit.The proposed approach accepts simulated natural fermionic noise as an alternative to demanding complete suppression of physical errors.Source: §1 · pp. 3–4

2. Preliminaries

Fermionic lattice models map discrete fermionic modes to qubits, while the Verstraete-Cirac and Low-Weight encodings use stabilizer constructions to make local lattice operators low-weight. Their first-order undetectable noise is predominantly local fermionic dephasing, linking encoding design to physically natural noise.

2.1. Fermionic Lattice Models

  • Fermionic lattice models represent discrete modes arranged on a lattice with short-range interactions, including the Fermi-Hubbard model.The Fermi-Hubbard model describes spin-1/2 fermions hopping on a square lattice with on-site interactions.Source: §2.1 · pp. 4–5
  • Majorana operators are self-inverse, mutually anticommuting, and flip fermion parity when applied.These properties provide the operator language used by both encodings.Source: §2.1 · p. 5

2.2. The Verstraete-Cirac Encoding

  • The Verstraete-Cirac encoding uses auxiliary fermionic sites and paired auxiliary-Majorana stabilizers to cancel Jordan-Wigner strings for nonconsecutive neighboring sites.The auxiliary sites alternate with primary sites in the ordering, and the pairing construction requires lattice degree at most four.Source: §2.2 · p. 6
  • The VC construction makes local fermionic operators geometrically local by multiplying long-string Jordan-Wigner representatives by stabilizers.This produces logically equivalent operators without long-range Z strings.Source: §2.2 · pp. 6–7

2.3. The Low-Weight Encoding

  • The Low-Weight encoding places qubits on lattice vertices and odd faces, orients edges, and represents fermionic edge operators with local Pauli operators.Boundary edges omit the Pauli operator associated with a nonexistent face qubit.Source: §2.3 · pp. 7–8
  • In the Low-Weight encoding, fermionic loop operators become stabilizers, with a cycle basis providing sufficient stabilizer generators.For square lattices, even-face loops yield explicit stabilizers involving vertex and face qubits.Source: §2.3 · pp. 8–9

2.4. Natural Noise on Fermionic Lattice Models

  • Phonon-induced momentum-space hopping appears as position-basis fermionic dephasing, providing a natural noise model for lattice fermions.The paper derives this effective noise model more fully in Appendix A.Source: §2.4 · p. 10
  • For both VC and Low-Weight encodings, first-order noise is dominated by natural fermionic noise, sometimes exclusively.This supports error mitigation through the physical interpretation of undetectable errors rather than only through high code distance.Source: §2.4 · p. 10

3. Mapping Physical Errors to Logical Errors

The paper classifies single-qubit errors in the VC and LW encodings, finding that most undetectable errors become local fermionic phase noise while rare Majorana errors can switch parity. It then proposes encoding modifications, parity checks, and randomized correction to mitigate these edge cases.

3. Mapping Physical Errors to Logical Errors

  • Most undetectable weight-1 errors in both encodings become low-mode-weight, local fermionic noise, with detectable errors and rare Majorana operators forming the remaining categories.The analysis assumes weight-1 Pauli noise dominates.Source: §3 · p. 10

3.3. Mitigating Parity Switching Errors

  • Parity-switching Majorana errors can violate fermionic parity superselection, motivating targeted mitigation rather than treating all undetectable errors as benign.A global parity stabilizer can detect these errors but may be costly for coherent, nondestructive measurements.Source: §3 · p. 10, §3.3 · p. 13

3.1. Verstraete-Cirac Encoding

  • In the VC encoding, undetectable errors act on primary fermionic sites or are stabilizers; nontrivial cases are primary-qubit Z errors and X/Y errors on the first ordered qubit.Primary Z errors map to single-mode phase noise, whereas the first-site X/Y errors map to single Majorana operators.Source: §3.1 · pp. 11–12

3.2. LW Encoding

  • In the LW encoding, face-qubit errors are detectable, while vertex-qubit Z errors yield phase noise and corner X/Y errors adjacent only to odd faces yield Majorana or hole operators.The corner Majorana cases are parity-switching errors and are excluded when only the even fermionic subspace is encoded.Source: §3.2 · pp. 12–13

3.3.1. Verstraete-Cirac Encoding

  • For the VC encoding, swapping the first primary and auxiliary sites makes first-site Majorana errors weight-2 without increasing local interaction-term weight.The modified ordering begins 1′, 1, 2, 2′, 3, 3′, … .Source: §3.3.1 · p. 14

3.3.2. LW Encoding

  • For the LW encoding, an even-by-even lattice can avoid parity-switching errors, while removing corner sites makes remaining corner Majorana or hole operators weight-2.The lattice modification adds a diagonal edge operator to preserve the required anticommutation structure.Source: §3.3.2 · p. 14

3.4. Partial Correction of Detectable X and Y Errors

  • Randomly correcting indistinguishable primary-site X/Y syndromes succeeds with 50% probability; otherwise it introduces an undetectable Z phase error, enabling active correction with added fermionic phase noise.All single-qubit Pauli errors on auxiliary qubits are distinguishable in both encodings.Source: §3.4 · pp. 14–15

4. Periodic Boundary Conditions

Periodic boundaries regularize the LW encoding’s error behavior: on suitable even tori, weight-1 and weight-2 errors are either detectable or correspond to natural fermionic phase noise, while boundary effects create exceptions.

  • Boundary effects introduce weight-1 Majorana or Majorana-hole operators and allow stabilizer violations from multiple errors to cancel.Source: §4 · p. 15
  • The periodic construction applies when the lattice has an even number of faces, equal to vertices, in both directions.Source: §4 · p. 15
  • On an even torus with at least four faces in each direction, any two weight-1 Pauli errors are detectable, or reduce to one detectable error plus phase noise, or two phase-noise terms.Source: §4 · p. 15
  • Periodic boundaries eliminate truncated boundary syndromes, leaving only X–Y cancellations on one vertex, which become a Z error and hence fermionic phase noise.Source: §4 · p. 16
  • For sufficiently large periodic lattices, all weight-2 errors are either detectable or map to natural simulated noise; without periodicity, non-phase undetectable errors are confined to boundaries.Source: §4 · p. 16

5. Outlook and Discussion

The discussion connects detectable-error post-selection and natural-noise mappings to near-term fermionic simulation, while identifying higher-weight errors and hardware bias as important directions and conditions.

  • In NISQ simulations, stabilizer measurements can support post-selection without adding full error correction, but coherent evolution must remain short enough that error probabilities stay ≪1.Source: §5 · p. 16
  • Spatially separated single-qubit errors can be detected between Trotter steps, whereas pairs whose syndromes cancel remain undetectable and limit accuracy.Source: §5 · p. 16
  • Higher-weight error mappings remain an open direction; if they produce natural local fermionic noise, mitigation could extend to higher orders.Source: §5 · pp. 16–17
  • In the LW encoding, undetectable weight-2 errors correspond to boundary terms, mostly paired hopping terms, suggesting coupling the simulated boundary to a superconducting system.Source: §5 · p. 17
  • The encodings preferentially tolerate particular weight-1 Pauli errors, making them especially relevant when hardware noise is already Pauli-biased.Source: §5 · p. 17
  • For fermion-density measurements, stabilizers can be measured destructively in the Z basis, allowing detection of Majorana errors without modifying the encoding.Source: §5 · p. 17
  • Mitigating many errors before error correction could reduce overhead even in fault-tolerant architectures.Source: §5 · p. 17

A. Derivation of Fermionic Phase Noise

The paper derives fermionic phase noise from a fermion–boson lattice interaction, using a thermal bosonic bath and a simplified momentum-independent coupling to obtain an effective spin-up channel.

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