Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 342 (Topological Quantum Matter)

Represent quasiparticle worldlines by conserved currents $j_I^{\alpha }$ and add the minimal-coupling termUp to the corresponding sign convention for charge, variation of the Abelian Chern--Simons theory givesFor a static quasiparticle of integer gauge-charge vector $q$, integration over a small surrounding disk yields the Chern--Simons flux attachment relationA quasiparticle of charge $q^{\prime }$ carried once around this flux acquires the Aharonov--Bohm phaseso

The constituent-particle current has charge vector $t=(1,\dots ,1{)}^T$. Its integral over the same disk isFor $K=I_n$ and $q=e_J$, this gives $N=1$. The self-exchange angle is ${\theta }_{qq}=\pi e_J^T{e}_J=\pi$, so the excitation is a fermion with one unit of microscopic particle number and may be identified with an electron.

Since $M^2=nM$, direct multiplication gives the matrix inverseprovided $1+np\ne 0$, as required by invertibility of the K-matrix. For $q=e_J$,The attached gauge-flux vector is thereforeRelative to the $K=I_n$ electron in part a, every one of the $n$ gauge fields consequently carries the additional flux

Because $t^{T}M=n{t}^T$,Thus a fundamental quasiparticle has fractional constituent number

Let $q$ be the gauge-charge vector of a local electron. Trivial full braiding with every integer quasiparticle $q^{\prime }$ requiresEquivalently $K^{-1}q=\ell$ for some $\ell \in {\mathbb{Z}}^n$, so the allowed local charges lie in the local-particle latticeUnit constituent number requiresFinally its self-exchange angle isFor integer ${\ell }_I$, one has ${\ell }_I^2\equiv {\ell }_I(\mathrm{mod}2)$, and henceThe exchange phase is fermionic precisely when this integer is odd. Therefore

Let $x$ and $y$ denote the two oriented noncontractible cycles of the torus. Their transport operators intersect once, so the mutual braiding phase gives the Wilson-loop algebra of an Abelian Chern--Simons theoryThe integer charge vectors are identified modulo local electrons: $q$ and $q+K\ell$ have identical braiding with every quasiparticle. The distinct topological charges therefore form the finite quotient group

Choose a ground state that diagonalizes the mutually commuting $x$-cycle operators. Acting with $T_y(q)$ multiplies their eigenvalues by the character $q^{\prime }\mapsto e^{2\pi i{q}^{\prime T}{K}^{-1}q}$. This pairing is nondegenerate on $\mathcal{A}$: if the character is trivial for every $q^{\prime }$, then $K^{-1}q$ is integral and $q$ is the trivial class. Thus the $|\mathcal{A}|$ states obtained from inequivalent $q$ have distinct loop eigenvalues and are orthogonal. The torus ground-state degeneracy of an Abelian Chern--Simons theory is consequently at least $|\det K|$.

The all-ones matrix has eigenvalue $n$ along $t$ and eigenvalue zero on the $(n-1)$-dimensional perpendicular subspace. Hence $K=I_n+pM$ has eigenvalues $1+np,1,\dots ,1$, and

The commuting termsare independent stabilizer generators. Since $J>0$, every ground state has $S_j=+1$. The resulting two-dimensional stabilizer subspace isthe phase-flip repetition code. A convenient pair of logical Pauli operators isIndeed, they commute with every stabilizer, anticommute with each other, and are not stabilizers.

For phase-flip errors $E_A={\prod }_{j\in A}{Z}_j$ and $E_B={\prod }_{j\in B}{Z}_j$, the operator entering the Knill--Laflamme condition is $E_A^{†}{E}_B=E_{A△B}$ and has weight at most $2t$. Every nonempty proper product of $Z_j$ anticommutes with some $S_j$ unless it is the full logical operator $\overline{X}$. The Knill--Laflamme conditions therefore hold whenever $2t<n$, and fail once two allowed errors can differ by $\overline{X}$. Thus

Each physical $X_j$ commutes with all stabilizers, and $X_j{X}_1$ is a product of stabilizers. Hence every $X_j$ acts on the code as the undetectable logical operator $\overline{Z}$. The code cannot detect, and therefore cannot correct, even a single bit flip.

A syndrome determines a phase-error set only up to multiplication by $\overline{X}=Z_1\cdots Z_n$: an error of weight $w$ and its complement of weight $n-w$ have the same error syndrome. Their probability ratio isFor $p<1/2$, maximum likelihood therefore chooses the representative of smaller weight. This is exactly majority-vote decoding of a repetition code: correction succeeds when fewer than half the qubits are flipped, with a random tie-break at $w=n/2$ when $n$ is even.

If $W\sim \mathrm{Bin}(n,p)$, then $\mathbb{E}W=np$ and $\mathrm{sd}(W)=\sqrt{np(1-p)}$. For every fixed $p<1/2$, the distance from the mean to the decision boundary in standard deviations isThus the logical failure probability $\mathrm{Pr}(W>n/2)$ tends to zero, in fact exponentially by a Chernoff bound. At $p=1/2$ the two representatives are equiprobable and decoding cannot improve with $n$. Hence

Label the physical qubits by $(i,j)$, with outer block $1\le i\le n$ and inner position $1\le j\le m$. The inner bit-flip repetition code has stabilizersand logical operators ${\overline{X}}_i={\prod }_{j=1}^m{X}_{i,j}$ and ${\overline{Z}}_i=Z_{i,1}$. Replacing each outer $X_i$ by ${\overline{X}}_i$ gives the outer stabilizers

This is the surface code on a chain of spheres. The $n+1$ pole-touching points are vertices, and the $m$ longitudes on sphere $i$ are its qubit-carrying links. At an interior touching point, the star operator is exactly $S_i^X$. Each face between adjacent longitudes has the two-edge plaquette operator $Z_{i,j}{Z}_{i,j+1}$; only $m-1$ of the $m$ face operators on each sphere are independent.

Using the conventions of part a, logical operators areThe first may be placed on any one sphere and is a dual equatorial cut crossing all $m$ longitude links. The second may use any fixed longitude and is a pole-to-pole path through all $n$ spheres. Multiplication by stabilizers deforms either representative without changing its logical action. Their minimum weights are $m$ and $n$, respectively, so the distance of a stabilizer code is

Under independent physical phase flips, an inner block suffers an effective phase flip of its encoded qubit exactly when it contains an odd number of errors. The binomial parity identity givesand hence the effective outer error probability isFor every $p<1/2$ and finite $m$, one has $q_m<1/2$, so outer majority-vote decoding succeeds with probability tending to one as $n\to \infty$. At $p=1/2$, one has $q_m=1/2$. The physical threshold is therefore still

For fixed $p\in (0,1/2)$, however, $q_m$ increases monotonically to $1/2$ as $m$ grows. Its distance from threshold isConsequently the majority-vote standardized separation is only of order $(1-2p{)}^m\sqrt{n}$, and the large-deviation exponent is of order $n(1-2p{)}^{2m}$. Thus larger $m$ leaves the threshold unchanged but makes the logical phase-error probability decay more slowly with $n$.

The continuum Hamiltonian is that of a spinless one-dimensional p-wave topological superconductor. Its topological phase has $\mu >0$, whereas the empty vacuum may be modeled as the trivial phase with $\mu <0$. With superconductor at $x<0$ and vacuum at $x>0$, this is

Near the transition $\mu =0$, momenta are small and the quadratic term may be neglected. In Pauli-matrix notation, the long-wavelength Bogoliubov--de Gennes Hamiltonian isThe zero-energy equation becomesChoose a constant spinor ${\chi }_{+}$ with ${\tau }_y{\chi }_{+}={\chi }_{+}$. Thendecays on both sides because $\mu$ changes from positive to negative. For asymptotically constant $|\mu |$, the Continuum p-wave Majorana interface mode has widthIts characteristic momentum is $|p|\sim |\mu |/\Delta$. The neglected kinetic energy is small compared with $|\mu |$ when $|\mu |\ll m{\Delta }^2$ up to a factor of two, equivalently when $\xi \gg \hslash /(m\Delta )$.

In the Nambu basis, Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian lets a zero-energy eigenvector be chosen self-conjugate, $v(x)=u(x{)}^{\ast }$. The corresponding quasiparticle operator isTaking the adjoint and interchanging the two terms gives ${\gamma }^{†}=\gamma$. With the usual normalization it obeys $\{\gamma ,\gamma \}=2$, so it is a Majorana fermion operator localized at the interface.

A Majorana zero mode is a localized Hermitian quasiparticle operator ${\gamma }_j={\gamma }_j^{†}$ that commutes with the Hamiltonian within the low-energy approximation and obeys the Clifford relationsAn exchange is adiabatic when its duration is long compared with $\hslash$ divided by the bulk excitation gap, so the adiabatic theorem keeps the state in the ground-state manifold. It should also be short compared with the inverse exponentially small zero-mode splitting if that splitting is not neglected.

Locality and conservation of fermion parity imply that exchanging modes $a$ and $b$ mapswith all other well-separated Majoranas unchanged. Since $({\gamma }_a{\gamma }_b{)}^2=-1$, exponentiation givesDirect conjugation using the Clifford relations produces exactly the displayed quarter-turn of $({\gamma }_a,{\gamma }_b)$. Thus, up to an overall physically irrelevant phase,The two signs are inverse Majorana braiding operators and correspond to the two braid orientations, clockwise and counterclockwise.

For $a\ne b$, the operator $Q_{ab}=i{\gamma }_a{\gamma }_b$ is Hermitian andThereforeFor $f=({\gamma }_a+i{\gamma }_b)/2$, one has $Q_{ab}=2f^{†}f-1$, so these projectors distinguish the two occupations, equivalently the two values of pair fermion parity. This proves thatdescribe a fermion-parity measurement.

Let $P={\Pi }_{+}^{(N0)}$ and suppose $P|\psi ⟩=|\psi ⟩$. The bilinears $i{\gamma }_a{\gamma }_0$ and $i{\gamma }_b{\gamma }_0$ each anticommute with $i{\gamma }_N{\gamma }_0$, whereas their product commutes with it. Expanding the projectors and sandwiching by $P$ therefore givesAfter normalizing the post-measurement state, the sequence consequently implementson the encoded ground space. This is Measurement-only Majorana braiding.

Immediately before measuring $Q_{b0}=i{\gamma }_b{\gamma }_0$, the state has definite parity for a bilinear such as $Q_{a0}$ or the restored reference $Q_{N0}$ that anticommutes with $Q_{b0}$. If $Q|\psi ⟩=s|\psi ⟩$ and $\{Q,Q_{b0}\}=0$, thenHence

If the undesired value of $s_b$ occurs, perform a reference measurement that projects the ancillary pair back toward its previous parity sector, then measure $Q_{b0}$ again. Alternating these anticommuting parity measurements gives a fresh probability $1/2$ of the desired result on each attempt. This Forced Majorana parity measurement has a geometric waiting time, succeeds almost surely, and does not measure the encoded parity $i{\gamma }_a{\gamma }_b$ directly. Apply the same repeat-until-success procedure to each required outcome in ${\Pi }_{+}^{(N0)}{\Pi }_{s_a}^{(a0)}{\Pi }_{s_b}^{(b0)}$; alternatively, keep arbitrary outcomes and track the known inverse-braid or Pauli-frame correction determined by their signs.