Bosonic Codes from Compact Phase Spaces

A quantum code protects information by picking out a subspace and asking that certain operators leave it alone. The Gottesman–Kitaev–Preskill code does this with one bosonic mode and a lattice of displacements: shift the oscillator by a lattice vector and nothing changes. Quotient the phase plane by that lattice and you get a torus. The code words turn out to be theta functions on it, and because the torus is compact the code space is finite-dimensional, with a dimension the Riemann–Roch theorem knows.

That last sentence is a geometer’s sentence, and it invites a geometer’s question. A torus is a surface of genus one. What code lives on a surface of genus two?

A phase space with a shape

The construction needs one thing of the phase space: a family of coherent states attached to its points, varying holomorphically. On the plane that is the usual Bargmann family, and the overlap of a state with the coherent state at is an entire function whose zeros say how far the state is from Gaussian. On a compact surface no such function exists globally — the only holomorphic functions on a compact surface are constants — so the overlaps live on patches and glue into a section of a line bundle.

Compactness then makes the picture rigid in a way the plane never is. Every state has the same number of zeros, that number is a topological invariant, and two states with the same zeros are the same state up to a scalar. Every state is pinned by a finite constellation of points on the surface.

For physicists

For stellar rank , Riemann–Roch gives

with the area of the surface — the Bohr–Sommerfeld rule, written for a phase space with topology. The genus-zero case is the Majorana constellation of a spin; the plane is the stellar hierarchy of a bosonic mode; this is the same statement for everything in between.

Eq. (1) · equation numbers throughout are those of arXiv:2608.31156v1

Squeezing instead of shifting

Every compact surface of genus two or more is the Poincaré disk folded up by a discrete group. That is fortunate, because the disk is a phase space a laboratory can build: two bosonic modes carry a representation of , and the points of the disk are the two-mode squeezed translates of a vacuum. The symmetries of the disk act as Gaussian operations.

So take the group that folds up the surface, and impose its elements as stabilizers. Where GKP asks a state to be unchanged by a displacement, this asks it to be unchanged by a squeeze. A state that survives all of them descends to a holomorphic differential on the surface, and the dimension of the code space follows again from Riemann–Roch.

For physicists

for , with a cocompact Fuchsian group and the weight, which fixes the sector of fixed occupation-number difference. The stellar function of a code state transforms as a holomorphic -differential, so the code space is the space of sections of , of degree . At and genus two the code space is two-dimensional: a logical qubit.

Eqs. (3)–(4)

The octagon

The simplest genus-two surface is the Bolza curve: a regular hyperbolic octagon with opposite sides glued. Four gluings, and each one is a two-mode squeezing operator of the same strength — the four stabilizer generators of the code, at of squeezing, in four directions a right angle’s eighth apart.

Getting the code words is a problem in classical geometry. Cut the surface along closed geodesics into pairs of pants — genus two takes exactly three cuts — and each geodesic supplies a theta series. Those three series span the weight-two code space. Then the whole tower of higher weights follows from one algebraic condition: find the combination in which the three satisfy a conic relation, take square roots, and every code word at every weight is a product of the two roots and their Wronskian.

For physicists

one series per conjugacy class of hyperbolic elements, one class per closed geodesic. For Bolza the three shortest geodesics give , and the change of basis into the conic form is fixed by a single complex number, . Then and are single-valued on the disk and form the weight-one basis; the products , together with the Wronskian times lower products, give a basis at every . The averaging that builds GKP code words diverges at , which is why the weight-one basis has to be reached from weight two rather than constructed directly.

Eqs. (7)–(9)

Any finite group you like

Here is what the curvature buys. Every symmetry of the surface is a logical gate: it permutes the stabilizers and preserves the code space, and because it is a Möbius transformation of the disk, it is a Gaussian circuit on two modes — a rotation, or a squeeze. The Bolza curve has forty-eight symmetries, and they are the single-qubit Clifford group.

By Greenberg’s theorem every finite group is the automorphism group of some compact Riemann surface. So every finite group is available as the logical gate group of one of these codes — in two bosonic modes, with Gaussian operations only.

At Bolza the order-eight rotation and an order-three Möbius map generate ; the second is three octagon rotations followed by a stabilizer squeeze at half strength, about 6.6 dB of two-mode squeezing. Away from integer weight the symmetries act only projectively and only a subgroup survives.

And then it does not work

A code needs a state. Average the stabilizers and you get the Markov operator of a random walk on the group; the stabilizer Hamiltonian is one minus that operator, and its lowest eigenvalue says how close a normalizable state can come to satisfying every stabilizer at once. For GKP that lowest eigenvalue is zero, which is exactly why approximate GKP code words of arbitrary precision exist. For genus two or more it is not zero, and it cannot be made zero.

The reason is a property of the group and nothing else. A group is amenable when a random walk on it comes home often enough; is, and the surface groups of genus two and up are not. There is simply too much room in a hyperbolic surface for the walk to find its way back.

genus 2 spectral radius 0.661 gap 0.339

Too much room to come home. The chance that a symmetric random walk started at the identity is back there after steps, on a logarithmic axis. On — the group behind GKP — it is exactly , which falls off like : a power law, spectral radius one, no gap. On a genus- surface group there are generators and no abelian relations to help; the free group on the same generators gives the limiting case, a walk on a -regular tree, whose return probability falls off exponentially with spectral radius . The curve leaves the bottom of the frame; the gap is what is left over.

For physicists

The theorem is a chain of three inequalities: . The last is Kesten’s theorem — every cocompact Fuchsian group is non-amenable, so the walk in the regular representation is gapped. The middle one is weak containment: the representation on is weakly contained in the regular one, and weak containment never raises an operator norm. So the gap transfers from the group to the code.

Quantitatively, at genus two Nagnibeda’s bound gives for every code on the standard four generators, since it sees only the abstract group. The free group on those four generators gives from the other side, and the surface group is a quotient of it, so the two nearly meet. For the Bolza surface at the lowest eigenvalue of , extrapolated from truncation, is about : three quarters of it is the group alone.

Eq. (11) and Theorem 1

Three geometries, then, and one dividing line running between the second and the third.

genus zero · positive curvature The sphere

The stabilizer group is finite. Nothing to converge, nothing to diverge.

exact code words exist

genus one · flat The torus — GKP

The stabilizer group is : infinite, but amenable. The stabilizer Hamiltonian’s spectrum reaches down to zero.

approximate code words, to any precision

genus two and up · negative curvature The hyperbolic surface

The surface group is non-amenable, so the spectrum stops short of zero. No normalizable state satisfies the stabilizers, and none comes close.

no code words, not even approximate ones

What could still be built

The obstruction is a statement about exact unitary stabilizers, and that leaves a door open. The gates themselves are ordinary circuit quantum electrodynamics: two microwave cavities coupled to a transmon, one parametric pump for the two-mode squeezing, and a cross-Kerr term for the correction. One stabilization round is a controlled squeeze, a measurement, and a conditional kick.

With a single generator it works: the stabilizer expectation climbs from 0.03 to 0.97 within five hundred rounds, from any starting state including the completely mixed one. Add a second generator along a different axis and the same protocol stops dead — not slower, stopped, at a plateau set by the gap. The number the simulation settles at is the theorem, arrived at from the other end.

The open question the paper leaves is whether finite-energy stabilizers — the unitaries damped by an envelope, as in the dissipative schemes that make GKP practical — can be made to tame a non-amenable geometry. That would be a way to keep the gates and pay for them differently.

Curvature is a recurring instrument elsewhere on this board: in Geometric Delocalization in Two Dimensions it decides whether a random walker on a surface ever comes home — the same question this paper asks of a walk on a group.