Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 (Topological Quantum Matter)

At a boundary parallel to lattice links, define each star as the product over links of the retained lattice incident on the vertex,A boundary vertex has three rather than four retained incident links. For a plaquette adjacent to the boundary, retainincluding its boundary link. A boundary link belongs to only one bulk plaquette, so applying $X$ on that link flips one $B_p$ rather than two. A magnetic string made from $X$ operators can therefore terminate at the boundary and its endpoint can be created or removed by a local boundary operator. This is precisely an $m$-condensing, or magnetic, boundary. By contrast, the same local operation does not permit an isolated electric endpoint.

With identical $m$-condensing boundaries, the cylinder supports one logical qubit and hence hasOne logical operator is an electric $Z$ string around the circumference; its conjugate is a magnetic $X$ string joining the two boundaries. The perturbation $h{\sum }_j{X}_j$ can generate the latter only after a virtual magnetic anyon traverses the length of the cylinder. Degenerate perturbation theory therefore gives the ground-state splitting of a surface-code cylinderwhere the factor $C$ counts translated shortest paths and nonuniversal order-one factors have been suppressed.

If both boundaries instead condense $e$, the unperturbed degeneracy remains two. The logical operator made solely from $X$ is now a magnetic loop winding around the circumference, so the same perturbation first acts nontrivially at order $C$:Thus exchanging the condensed anyon exchanges the geometrical length controlling this perturbative splitting.

The block matrix isIf both $q\in \mathcal{L}$ and $q^{\prime }$ condense, choose their boundary excursions so that the two string operators cross once. Their commutator is their mutual full-braiding phase:Both operators act as the identity on every ground state, so consistency requiresWriting $q^{\prime }=(q_{\mathrm{top}}^{\prime },q_{\mathrm{bot}}^{\prime })$, this saysSet $l_{\mathrm{top}}=M^{-1}{q}_{\mathrm{bot}}^{\prime }$ and choose $l_{\mathrm{bot}}=0$. Then $Kl=(0,q_{\mathrm{bot}}^{\prime }{)}^T$, andTherefore every additionally condensable anyon has the formThe $Kl$ term is a local particle in the anyon lattice of an Abelian Chern--Simons theory, so the condensate is maximal modulo local excitations, as required for a Lagrangian subgroup of Abelian anyons.

Since $\det M=-3$, the condensed top-sector anyons modulo local particles formwhich has order $|\det M|=3$. The same lower bound follows directly from Wilson-operator algebra. TakeBecausetheir crossing operators obeyActing repeatedly with one operator on an eigenstate of the other produces three states with distinct eigenvalues; they are linearly independent and have the same energy. HenceFor these identical maximal boundaries the bound is saturated, although only the lower bound was requested.

A recovery operation must reverse every coherent superposition of the errors ${\tilde{E}}_a$ without learning or disturbing the encoded state. Thus the corrupted subspaces associated with distinguishable syndromes must be orthogonal, while errors with the same syndrome must have identical action on the logical information. For any code states $|\psi ⟩,|\varphi ⟩$, this meanswith coefficients independent of the encoded states. In projector form these are exactly the Knill--Laflamme conditionsDiagonalizing the positive matrix $c_{ab}$ chooses error combinations with mutually orthogonal syndrome spaces, which can be measured and reversed without revealing logical amplitudes.

The centralizer of a stabilizer group isThe nontrivial logical Pauli operators are represented byor more invariantly by the quotient $C(\mathcal{S})/\mathcal{S}$ after phases are removed.

For $P=E_a^{†}{E}_b$, there are three cases. If $P\in \mathcal{S}$, it acts as a scalar on the code. If $P\notin C(\mathcal{S})$, it anticommutes with a stabilizer and ${\Pi }_{\mathcal{L}}P{\Pi }_{\mathcal{L}}=0$. If $P\in C(\mathcal{S})\setminus \mathcal{S}$, it acts as a nontrivial logical operator and is not proportional to the identity. Therefore

If every $\mathrm{wt}(E_a)<d/2$, thenBy the definition of the distance of a stabilizer code, no Pauli operator of weight below $d$ is a nontrivial logical operator. Part (b) therefore proves the local correctability of a stabilizer code for this error set.

The converse fails because pairwise products, rather than individual weights, control correctability. For example, let $P$ be a high-weight Pauli outside $C(\mathcal{S})$ and take the error set $\{I,P\}$. If $P$ anticommutes with a stabilizer, then $\Pi P\Pi =0$, while $P^{†}P=I$; the KL conditions hold even when $\mathrm{wt}(P)\ge d/2$. A still simpler singleton set containing any known unitary Pauli error is always reversible regardless of its weight.

For Pauli errors, “suitably local” means that every pairwise product has support too small to carry a logical string:A sufficient geometric statement is $|\mathrm{supp}{E}_a\cup \mathrm{supp}{E}_b|<d$; in particular, each error having weight below $d/2$ suffices. Such a product is either a stabilizer or creates an excitation detected by at least one stabilizer. It is never a nontrivial logical operator, so the ground-space projector of the surface code obeys the Knill--Laflamme condition for the entire set.

For sufficiently weak local fields the bulk gap remains open, so quasi-adiabatic continuation supplies a quasi-local unitary $U$ mapping the unperturbed ground space to the span of the $2^k$ lowest eigenstates:If $S_j$, $j=1,\dots ,n-k$, are independent original stabilizer generators, defineThen ${\tilde{S}}_j|{\phi }_{\alpha }⟩=|{\phi }_{\alpha }⟩$, and independence is preserved by conjugation. The operators ${\tilde{S}}_j$ are quasilocal, with exponentially decaying tails, but a generic perturbation makes them non-Pauli; this is the dressed stabilizer under a weak local perturbation.

Exact error correction is transported with the code: the exactly correctable dressed errors are $U{E}_a{U}^{†}$. A bare Pauli error is generally not one of these dressed operators. Its expansion in the dressed algebra has exponentially small long-range components that can act within the logical space, soThus a set of bare Pauli errors satisfies the KL conditions only approximately, with deviations suppressed exponentially by the code distance relative to the dressing length, except at specially tuned points.

Fermion parity is $P_f=(-1{)}^F$. An operator $O_{+}$ is parity even when $P_f{O}_{+}{P}_f=O_{+}$, equivalently $[P_f,O_{+}]=0$; it is parity odd when $P_f{O}_{-}{P}_f=-O_{-}$, equivalently $\{P_f,O_{-}\}=0$.

Odd operators supported in disjoint spacelike regions anticommute by the canonical anticommutation relations. If each were an observable, their measurement algebras would fail to commute, allowing the order of spacelike separated measurements to affect predictions. Products of an even number of fermionic fields instead commute at spacelike separation. Locality and compatible spacelike measurements therefore require every physical local observable to be fermion-parity even; parity-odd fields can create charged states but are not themselves observables.

The Bogoliubov--de Gennes Hamiltonian has particle-hole symmetry. A locally nondegenerate zero mode can therefore be chosen particle-hole invariant. If its Nambu wavefunction is $(u_j,u_j^{\ast })$, defineThen ${\gamma }_j^{†}={\gamma }_j$. Exponential localization and $\ell /\xi \to \infty$ make distinct zero-mode wavefunctions orthogonal. Normalizing each one and using the fermionic canonical anticommutation relations gives

Pair the $2M$ Majorana zero modes into ordinary zero-energy fermionsTheir occupations produce $2^M$ states. Physical operations preserve total fermion parity, so choosing one parity sector imposes one binary constraint and leaves $2^{M-1}$ states. Thus Dense encoding with Majorana zero modes gives

A local observable is parity even by part (a). In the infinite-separation limit, any nontrivial parity-even product of zero-mode Majoranas that acts on the encoded information has support near at least two separated zero modes and cannot occur in a single local observable. The projection of $O$ onto the fixed-parity ground space is consequently scalar. For every orthonormal ground-space basis,This is Local indistinguishability of separated Majorana zero modes.

At finite but large $\ell /\xi$, zero-mode wavefunctions overlap by $O(e^{-\ell /\xi })$. The Hamiltonian acquires couplings $i{ϵ}_{ij}{\gamma }_i{\gamma }_j$, the ground-state degeneracy is split, and local observables gain logical matrix elements of the same exponential order:

Write $a={\gamma }_{2M}$ and $b={\gamma }_{2M-1}$. The ancilla condition $iab|\psi ⟩=|\psi ⟩$ implies $a|\psi ⟩=ib|\psi ⟩$ and lets every occurrence of $a$ on the state be replaced by $ib$. Define the two parity projectorsTheir factors square to one, so they are valid fermion-parity measurement projectors. Expanding $P_1{P}_2$, using the Majorana anticommutation relations, replacing $a$ with $ib$ on $|\psi ⟩$, and usinggivesThe proportionality absorbs the probability amplitude for obtaining the two displayed measurement outcomes. This realizes the four-Majorana phase gate using one braid and parity measurements.

Let ${{\rm Y}}_{123}=i{\gamma }_1{\gamma }_2{\gamma }_3$. Reversing three mutually anticommuting Majoranas changes sign, soMoving ${\gamma }_4$ through the three factors givesThus ${{\rm Y}}_{123}$ and ${\gamma }_4$ obey exactly the algebra of two Majorana fermion operators. Their exchange is implemented bywhich conjugates one into the other, up to the orientation sign, just like a Majorana braiding operator.

Ordinary braids permute the elementary ${\gamma }_j$ with signs. Part (d) implements $e^{i\pi {\gamma }_1{\gamma }_2{\gamma }_3{\gamma }_4/4}$ by braids and parity measurements, and conjugation by this unitary maps an elementary Majorana to a Hermitian cubic monomial of the other three. Repeating this operation grows or shrinks an odd monomial by two factors, while ordinary braids place the desired indices in the active positions. By induction, every Hermitian odd monomial in ${\gamma }_1,\dots ,{\gamma }_{2M-2}$ can be reached from ${\gamma }_1$, with factors of $i$ inserted according to its degree to make it Hermitian. Hence braids plus suitable fermion-parity measurements map, in their action on $|\psi ⟩$,