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Exact finite-volume toric-code gap

Statement

For the finite toric-code Hamiltonian $H_{L} = - J_{e} \sum_{q} A_{q} - J_{m} \sum_{f} B_{f}$ with $J_{e}, J_{m} > 0$ , the ground space is the four-dimensional toric code space and the ground energy is $E_{g} = - L^{2} (J_{e} + J_{m})$ . The exact spectral gap above it is $Δ_{L} = 4 \min (J_{e}, J_{m})$ .

Facts & Assumptions

Given: $L \geq 3$ , fixed couplings $J_{e}, J_{m} > 0$ , the Hamiltonian, the toric code space, and the joint-check syndrome labels.

[F1]

Finite toric-code Hamiltonian: $H_{L}$ is the finite sum $- J_{e} \sum_{q} A_{q} - J_{m} \sum_{f} B_{f}$ .

[F2]

Toric checks decompose into joint syndrome eigenspaces: the check eigenvalue sectors are an orthogonal decomposition of the Hilbert space; each defect parity is even.

[F3]

The periodic toric code encodes two qubits: the common $+ 1$ eigenspace has dimension four.

[F4]

Open toric strings have endpoint syndromes: a one-edge primal Z string creates exactly two e defects, and a one-edge dual X string creates exactly two m defects.

[F5]

Toric e and m syndromes: e and m defect counts are the numbers of star and plaquette checks with eigenvalue $- 1$ .

Proof

1.1givenF1F2F5algebra

On a joint-check eigenspace with $N_{e}$ negative star signs and $N_{m}$ negative plaquette signs, each star sum is $L^{2} - 2 N_{e}$ and each plaquette sum is $L^{2} - 2 N_{m}$ . By [F1] the Hamiltonian acts there by the scalar $- L^{2} (J_{e} + J_{m}) + 2 J_{e} N_{e} + 2 J_{m} N_{m}$ .

2.1F2F3step 1.1algebra

The all-positive sector is the code space by definition, is nonzero by [F3], and has energy $E_{g} = - L^{2} (J_{e} + J_{m})$ . Every other nonzero sector has at least one defect species present; by [F2] its defect count is even, hence at least two. Step 1.1 then places its energy at least $4 \min (J_{e}, J_{m})$ above $E_{g}$ .

3.1F1F4step 2.1algebra∎

If $J_{e} \leq J_{m}$ , choose a one-edge primal path and apply its Z string to a nonzero code vector. By [F4] the result is a nonzero joint eigenvector with exactly two e defects, so its energy is $E_{g} + 4 J_{e}$ . If $J_{m} < J_{e}$ , use a one-edge dual path and its X string instead; it yields a nonzero joint eigenvector with exactly two m defects and energy $E_{g} + 4 J_{m}$ . These states attain the lower bound from step 2.1, proving $Δ_{L} = 4 \min (J_{e}, J_{m})$ .

Depends on

Used by

A finite toric gap does not imply arbitrary perturbation stability Counterexample
L=3 torus checks and syndrome calculation Example
A toric logical string has constant energy barrier Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alexei Kitaev, Fault-tolerant quantum computation by anyons (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)