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Building manifolds from quantum codes

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Abstract

We give a procedure for “reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over \({\mathbb {Z}}\). Applying this procedure to chain complexes obtained by “lifting" recently developed quantum codes, which correspond to chain complexes over \({\mathbb {Z}}_2\), we construct the first examples of power law \({\mathbb {Z}}_2\) systolic freedom. As a result that may be of independent interest in graph theory, we give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph, such that each edge appears only polylogarithmically times in the basis. We use this result to trivialize the fundamental group of the manifold we construct.

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Notes

  1. Throughout, triangulation refers to a PL triangulation.

  2. The adjective “Whitehead” [W40] refers to his condition that all k-simplexes of the triangulation are uniformly biLipschitz equivalent to a Euclidean ball (of some radius) and that there is a uniform bound on the number of simplexes that any given simplex meets.

  3. The work constructs codes on n qubits with distances (up to polylogarithmic factors) \(n^{3/5}\), where the distance is the minimum weight of a nontrivial homology or cohomology representative. This would gives only \(\alpha =\frac{1}{5}\); however, these codes result from “distance balancing" codes with distances \(d_X=n^{1/2},d_Z=n^{3/4}\), up to polylogarithmic factors. Thus, before balancing, one gets \(\alpha =1/4\) up to logarithmic factors.

  4. We use the term “stabilizer" as shorthand for “stabilizer generator" throughout.

  5. Yet another attempt would be to start with a bounded degree tree graph, with one leaf per X-stabilizer and attach \(S^1\) to the leaves; this will typically not give feature \((*)\) when attaching 2-cells since they will need to “stretch across the tree" and there may be many cells attached to a given edge.

  6. A cell complex where cell attachments are, by maps which piecewise are smooth and of maximal rank, between a region on the sphere boundary and the cells to which it is attached.

  7. Technically we should add that the metric has a totally geodesic product collar on its boundary so that the metric on the double is \({\mathbb {C}}^\infty \).

  8. Famously, \(f_p\) fails to be a function at p where we interpret the image \(f_p(p)\) not to be a point but the entire \(S^{n-1}\). So, technically, \(f_p\) is a closed relation, not a function.

  9. Closed means compact and without boundary.

  10. This is a property of those specific codes, but in general nontrivial zeroth or second homology for that chain complex implies redundant stabilizers which can be dropped without changing the distance of the code.

  11. We thank the referee for pointing out that \(b_2(M)\) might not vanish.

  12. In detail, \(H_1(G)\) is Alexander dual to \(H^3(B^5\setminus G)\) which is Lefschetz dual to \(H_2(B^5\setminus G, \partial (B^5\setminus G))\), a basis of which is spanned by the cores of the 2-handles.

  13. If we wished here to work in the PL category Theorem 1.8.1 would tell us that there is only one PL homeomoprhisms type, \(S^4 \times S^7\), among the \(\{M_i\}\). This is because the ambiguity in the smooth structure disappears PL: all homotopy spheres in dimension \(\ge 5\) are PL homeomorphic to an \(S^n\) [S][Z].

  14. A homological code is a quantum code with qubits identified with cells of a particular dimension and X- and Z-stabilizers identified with cells of one lower and one higher dimension on some cellulation of a manifold.

  15. A simple cycle is a closed path on a graph that does not repeat any vertices. Every simple cycle defines a closed 1-chain with \({\mathbb {Z}}_2\) coefficients, taking a coefficient 1 on all edges in the cycle and 0 on other edges. However, not all closed 1-chains correspond to simple cycles. Sometimes in graph theory a simple cycle is just called a cycle, but we use the term simple cycle to avoid confusion.

  16. See [santalo76].

  17. One way to see this is to consider the unique great \(S^{j+1}\) containing s and \(S^j\). The visual image of \(S^j\) in this \(S^{j+1}\) is the total field (with multiplicity 1) and equal to the visual image of \(S^j\) from s in \(S_s^{n-1}\), in both cases it has exactly the same j-area as the source cycle \(S^j\).

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Acknowledgements

The first named author would like to thank Xin Zhou for discussions on the isoperimetric inequality appearing in “Appendix B”.

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Appendices

Appendix A: Cycle Bases and Decongestion Lemma

Given a graph G, a cycle basis is a set of simple cyclesFootnote 15 such that the corresponding 1-chains form a basis for \(H_1(G;{\mathbb {Z}}_2)\).

A weakly fundamental cycle basis is a cycle basis which can be linearly ordered so that every cycle contains at least one edge which does not appear in any later cycle in the basis (some authors reverse this order).

Weakly fundamental cycle bases have the following useful property, that we can trivialize the fundamental group by attaching 2-cells to each cycle in the basis. This is stronger than what we might have for an arbitrary cycle basis, where attaching such 2-cells might only kill first homology without trivializing the fundamental group.

Lemma A.0.1

Given a connected graph G, regarded as a 1-complex, any weakly fundamental cycle basis is actually a free basis for \(\pi _1(G)\). As a consequence attaching 2-cells to this basis produces a simply connected result.

Proof

Let the basis have k cycles. Order the cycles \(C_1,\ldots ,C_k\) of the basis so that each \(C_i\) contains some edge \(e_i\) not in any later cycle. \(G\setminus (e_1 \cup \ldots \cup e_k)\) must be a tree; if not, the \(\{C_i\}\) do not even span \(H_1(G;{\mathbb {Z}}_2)\). Crushing this tree to a point, G becomes a bouquet of circles, the circles bijective with the edges \(\{e_i\}\), exhibiting \(\{e_i\}\) as a free basis. \(\square \)

Our main result is the following decongestion lemma. We do not need the results in the second paragraph of this lemma in the rest of this paper, but we give them for completeness. This lemma shows the existence of a weakly fundamental cycle basis in which each edge appears only polylogarithmically many times. Other than MacLane’s “2-basis", the property of bounding the number of times an edge appears in a cycle basis does not seem to have been considered much before. [reich2014cycle] called a basis in which each edge appears at most p times a “p-basis", but we have not found any upper bounds on p in the literature.

Lemma A.0.2

Given any graph (possibly with multi-edges and self-edges) on V vertices and E edges, with the degree of each vertex bounded by O(1), there is a weakly fundamental cycle basis in which each edge appears in at most \(O(\log (V)^2)\) cycles in the basis.

Further, there is an efficient randomized algorithm, given in the proof of this lemma, to construct such a basis. Further, if we assign non-negative weights to each edge of the graph, then with probability \(\Omega (1)\) the algorithm returns a cycle basis of total weight bounded by \((\log (V))\) times the sum of edge weights of the graph.

Proof

The proof is based on the following algorithm A.

The algorithm A is recursive and randomized. We write A(G) to denote the cycle basis returned given graph G as input. If G has no edges, then \(A(G)=\emptyset \). Otherwise:

  1. 1.

    If graph G has at least one vertex with degree 1, let v be an arbitrary such degree-1 vertex and define \(G'\) to the be graph with that vertex removed (i.e., the subgraph induced by the set of vertices other than v). Return \(A(G')\).

  2. 2.

    Else, if graph G has at least one vertex v with degree 2, let v be an arbitrary such degree-2 vertex. We consider two cases. Case A: If v has a self-edge e , let C be the cycle consisting just of that self-edge. Let \(G'\) be the graph obtained by removing edge e from G, and return \(\{C\} \cup A(G')\). Case B: If v does not have a self-edge, then v has edges to two other vertices xy. Define \(G'\) to be the graph given by removing v from G and adding an edge (xy). Compute \(A(G')\). Then, for each cycle in \(A(G')\), replace every occurrence of edge (xy) with (xv), (vy) and return this as A(G).

  3. 3.

    Else, find a simple cycle C of length at most \(O(\log (V))\) in G. Let \(G'\) be the graph obtained from G by removing an edge of that cycle, choosing that edge uniformly at random. Return \(\{C\} \cup A(G')\).

Let us prove first that this algorithm is correct, in that it always returns a weakly fundamental cycle basis for G, and let us prove that indeed a cycle of length \(O(\log (V))\) does exist in case 3. Then we use this algorithm to prove the existence of a cycle basis where each edge appears at most \(O(\log (V)\log (E))\) times.

The proof that the algorithm returns a cycle basis is inductive: we assume it is correct for all graphs with \(V'<V\) vertices and for all graphs on V vertices with \(E'<E\) edges, and use that to prove it holds for all graphs with V vertices and E edges. The base cases \(V=0\) or \(V>0,E=0\) are trivial. Under this inductive assumption, it is immediate that the algorithm returns a cycle basis. Note that if item 1 holds, then v does not participate in any cycle, and so a cycle basis for \(G'\) gives a cycle basis for G. The cycle basis the algorithm returns is weakly fundamental since each time in case 2A or 3 that we add a cycle C to the basis, we remove an edge in C.

To bound the length of the simple cycle in item 3, note that by assumption all vertices have degree at least 3. Hence, starting from any given vertex there are \(>2^l\) paths of length l starting at that vertex so for \(l\sim \log (V)\) there must be two different paths with the same endpoint, implying a simple cycle of length \(O(\log (V))\).

Now we estimate how many times some edge e appears in a cycle basis returned by A(G). The algorithm is recursive, starting with graph G and defining a sequence of graphs \(G',G'',G''',\ldots \). Let \(G_0=G,G_1=G',G_2=G'',\ldots \). Note that when the algorithm constructs \(G_{j+1}\) from \(G_j\), the edge set of \(G_{j+1}\) is some subset of the edges of \(G_j\), possibly union with an extra edge in case 2B. We say that an edge f in \(G_{j+1}\) is the child of an edge e in \(G_j\) if f is the same as edge e or if f is obtained in case 2B, where \(e=(x,v)\) or \(e=(v,y)\) and \(f=(x,y)\). We say that an edge f in \(G_j\) is the descendant of an edge e in \(G=G_0\) if there is a sequence \(e_0=e,e_1,e_2,\ldots ,e_j=f\) with \(e_{k+1}\) the child of \(e_k\) for \(k=0,1,\ldots ,j-1\); note that this descendant is unique, if it exists. Not every edge e has a descendant in \(G_j\): some edges are removed. We say that an edge \(e\in G\) is removed on step j if e has a descendant in \(G_{j-1}\) but does not have a descendant in \(G_j\).

Then, for any edge \(e\in G\), the number of times that e appears in the cycle basis is equal to the number of times that the algorithm constructs a cycle C containing a descendant of e. If on the j-th step, for some edge e we construct a cycle containing a descendant of e, using case 2A of the algorithm, then e is removed on step j. Hence, the number of times that e appears in the cycle basis is bounded by one plus the number of times that the algorithm uses case 3 and constructs a cycle C containing a descendant of e.

However, each time case 3 occurs, the descendant is removed with probability \(\Omega (1/\log (V))\). Hence, the probability that an edge e appears in at least w such cycles is bounded by

$$\begin{aligned} (1-\Omega (1/\log (V))^w. \end{aligned}$$

For some \(w=O(\log (V) \log (E))\), this probability can be made strictly smaller than 1/2E, and so by a union bound, no such edge exists with probability \(\ge 1/2\).

Suppose the input graph G has no multi-edges or self-edges (this is the case that is relevant for our applications). Then, \(E=O(V^2)\); this is the worst case for a dense graph, though for our applications with a sparse G we have \(E=O(V)\). So \(\log (V) \log (E)=O(\log (V)^2)\).

If G has multi-edges or self-edges, we can still prove the claim of the lemma by applying a “pre-processing" before running the algorithm. This case is not needed for us, but we give it for completeness. First, remove self-edges and replace every multi-edge with a single edge to obtain some \({\tilde{G}}\) with V vertices and \({\tilde{E}}=O(V^2)\) edges. Construct a cycle basis for \({\tilde{G}}\) in which every edge appears at most \(O(\log (V)^2)\) times. If an edge in \({\tilde{G}}\) connects two vertices which have multiple edges between them in G, replace each occurence of that edge in the cycle basis with an arbitrary one of those multiple edges in G. Then, add every self-edge of G as a cycle to this cycle basis. Also, if any two vertices uv in G have multiple edges, labeled \(e_1,e_2,\ldots ,e_k\) for some k, add the cycle \(e_j e_{j+1}^{-1}\) for each \(j=1,\ldots ,k-1\) to this cycle basis. The result is a cycle basis for G with the given property.

The algorithm A that we have given is efficient, since it terminates after at most E recursive calls and each step can be done efficiently (one can efficiently find a shortest cycle in the graph by, for example, for each edge in turn, considering the graph with that edge removed and searching for a shortest path between the vertices which were endpoints of the edge). Each time the algorithm is run, it succeeds in finding such a basis with probability \(\ge 1/2\), and the desired properties of the basis can be efficiently verified.

Suppose non-negative edge weights are assigned to each edge. Then expected cycle basis weight is, by linearity of expectation, the sum over edges of the weight of that edge times the expected number of times the algorithm constructs a cycle C containing a descendant of the edge. That expected number of times is \(O(\log (V))\). Hence, the expected cycle basis weight is bounded by the total edge weight times \(O(\log (V))\), so with probability \(\Omega (1)\), the cycle basis weight is bounded by total edge weight times \(O(\log (V))\). \(\square \)

Appendix B: Pushing to Boundary: Alternative Proof

Here we give an alternative to lemma 1.6.3 that may be of independent interest as an isoperimetric inequality.

Lemma B.0.1

Let \((B^k,S^{k-1})\) be the unit ball in Euclidean k-space. Let V be a smooth singular p-cycle in \(S^{k-1}\) for \(p < k-1\). Suppose V is the boundary of a smooth singular chain W lying in \(B^k\), then V is also the boundary of a smooth singular chain \(\overline{W}\) lying in \(S^{k-1}\) with

$$\begin{aligned} p{{\,\mathrm{-area}\,}}(\overline{W}) \le 2^p(1+2\pi )p{{\,\mathrm{-area}\,}}(W). \end{aligned}$$

Proof

By the compactness properties of varifolds [federer], W may be replaced with a least area varifold Z, \(\partial Z = V\), and \(p{{\,\mathrm{-area}\,}}(Z) \le p{{\,\mathrm{-area}\,}}(W)\). By proposition 3.7 of [cm11], the monotonicity formula holds in the context of stationary varifolds. This formula tells us that if we divide Z into \(Z_{\le 1/2} \cup Z_{>1/2}\), the portions of Z inside and outside \(B^k_{1/2}\), the ball of radius \(\frac{1}{2}\), that

$$\begin{aligned} p{{\,\mathrm{-area}\,}}(Z_{>1/2}) > \left( 1 - \left( \frac{1}{2}\right) ^p\right) p{{\,\mathrm{-area}\,}}(Z). \end{aligned}$$
(B.0.1)

Furthermore the co-area formula tells us that

$$\begin{aligned} dp{{\,\mathrm{-area}\,}}(Z_{>1/2}) \ge \int _{\tau =1/2}^1 (p-1){{\,\mathrm{-area}\,}}(Z \cap S_\tau ^{d-1})\ d\tau . \end{aligned}$$
(B.0.2)

\(S_{\tau }^{d-1}\) the sphere of radius \(\tau \).

Lines B.0.1 and B.0.2 imply that for some \(\tau \in (\frac{1}{2},1)\) \((p-1)\)-area of \(Z \cap S_{\tau ^{d-1}}\) satisfies:

$$\begin{aligned} (p-1){{\,\mathrm{-area}\,}}(Z \cap S_\tau ^{d-1}) \le 2\left( 1 - \left( \frac{1}{2}\right) ^p\right) p{{\,\mathrm{-area}\,}}(Z) < 2p{{\,\mathrm{-area}\,}}(Z). \end{aligned}$$
(B.0.3)

Now let C be the cone \(C(Z \cap S_\tau ^{d-1})\) embedded in \(S_\tau ^{d-1}\) as follows. Carefully, as described next, choose a point \(\overline{q} \in S_\tau ^{d-1}\) disjoint from Z. Let C be the geodesic cone where each point of \(Z \cap S_\tau ^{d-1}\) is joined to q, the antipode of \(\overline{q}\), by the unique shortest geodesic arc (which has length \(<\pi \tau \)).

Lemma A

Given any piecewise smooth j-cycle and an appropriate choice of a point \(q \in S_\tau ^{d-1}\), then \(p{{\,\mathrm{-area}\,}}(C) < \pi \tau ((p-1){{\,\mathrm{-area}\,}}(Z \cap S^{d-1})) \le 2\pi \tau p{{\,\mathrm{-area}\,}}(Z)\).

We will need a small digression into integral geometry to prove Lemma A. For motivation recall Croften’s Theorem, that the length of a rectifiable curve \(\gamma \in {\mathbb {R}}^n\) is (up to a closure of a constant ) the integral over the kinematic measureFootnote 16 on \(\{l\}\), the lines of \({\mathbb {R}}^n\) of the number of intersection points, \(\left|l \cap \gamma \right|\)

$$\begin{aligned} L(\gamma ) \propto \int _{\{l\}} \left|l \cap \gamma \right|. \end{aligned}$$
(B.0.4)

An equivalent formulation is that

$$\begin{aligned} L(\gamma ) \propto \int _{r \in {\mathbb {R}}^n} d(\text {vol}) L(\gamma _r) \end{aligned}$$
(B.0.5)

where \(\gamma _r\) is the projection of \(\gamma \) into the visual sphere of oriented rays emanating from \(r \in {\mathbb {R}}^n\) and the length in the integral is w.r.t spherical geometry of \(S^{n-1}\).

In spherical geometry there is also the notion of the visual \(S_s^{n-1}\) sphere from each point \(s \in S^n\), defined by the oriented geodesic arcs of length \(\pi \) leaving s. Very similar to (B.0.5) is

Lemma B

Let \(\gamma \) be a piecewise smooth singular j-cycle in \(S^n\), \(j < n\), then

$$\begin{aligned} j{{\,\mathrm{-area}\,}}(\gamma ) = \int _{s \in S^n} d(\text {vol}) j{{\,\mathrm{-area}\,}}(\gamma _s) \end{aligned}$$

(actually the constant of proportionality is one in this case).

Proof

To develop some intuition, consider the appearance of a small round j-disk \(\Delta \) of radius \(= \delta > 0\) whose center is at some distance \(\Delta \) from s. One may check that the apparent area, i.e. the area in the visual sphere about s, is

$$\begin{aligned} {\text {A}}_s(\Delta ) = \Omega _j \left( \prod _{i=1}^j \sin (\beta _i)\right) /\sin (t) \end{aligned}$$
(B.0.6)

where \(\Omega _j\) is the j-volume of the unit j-ball, and where the \(\beta _i\) are the j dihedral angle between \(\Delta \) and the ray from s to the origin of \(\Delta \).

The important thing to notice is that \(\sin (t)\) in the denominator, objects very close to distance 0 or distance \(\pi \) look enormous.

We view every small speck of \(\gamma \), which we may think of as a tiny disk in a smooth singular simplex of \(\gamma \), so small that it is indeed well-approximated by a round \(j\text {-ball}_\delta \) such as \(\Delta \). So, it is very easy to believe that the only thing that, on average, could affect \(\int _{s \in S^n} {\text {A}}_s(\gamma )\) is the area \({\text {A}}(\gamma )\). This can be established by elementary approximation arguments and observing that both sides of (B.0.6) are additive under gluing. In fact, for every s

$$\begin{aligned} {\text {A}}_s(X \cup Y) = {\text {A}}_s(X) + {\text {A}}_s(Y) - {\text {A}}_s(X \cap Y) \end{aligned}$$
(B.0.7)

since these are just ordinary j-areas in \(S^{n-1}\).

To see that the constant is indeed unity for equation B.0.7, notice that if \(S^j \subset S^n\) is any totally geodesic (“great”) j-sphere in the n-sphere that from any point \(s \in S^n\) (disjoint from \(S^j\), or on \(S^j\)) the visual image of \(S^j\) in the visual sphere \(S_s^{n-1}\) is also a great \(S^j \subset S_s^{n-1}\).Footnote 17 This example shows that the constant is one: for a great \(S^j\), not just the average, but every individual visual image has the same area as the original j-cycle. This completes the proof of Lemma B. \(\square \)

From Lemma B we can now prove an isoperimetric inequality, Lemma C, whose statement and proof result from discussions with Xin Zhou. Lemma A is the immediate application of Lemma C.

Lemma C

Let \(\gamma \) be a piecewise smooth singular j-cycle \(\gamma \subset S^n\). There is a point \(q \in S^n\) so that the core \(C_q(\gamma )\) the cone of geodesic segments of length \(\le \pi \) beginning on \(\gamma \) and ending at a satisfies:

$$\begin{aligned} (j+1){{\,\mathrm{-area}\,}}(C_q(\gamma )) \le \pi (j{{\,\mathrm{-area}\,}}(\gamma )). \end{aligned}$$

Proof

Since \(\int _{s \in S^n} {\text {A}}(\gamma _s) = {\text {A}}(\gamma )\), where we have normalized so that \(\int _{S^n} 1 = 1\), there exists point \(q \in S^n\) so that \({\text {A}}(\gamma _q) \le {\text {A}}(\gamma )\). Now \(C_q(\gamma )\) contains a geodesic segment of length \(\le \pi \) for every point of \(\gamma \). In the Euclidean case Fubini’s then would conclude the proof, but the positive curvature, bending the fibers together reduces the area of the cone making the inequality hold even more strongly. \(\square \)

To translate Lemma B to Lemma C set \(n = d-1\) and \(j = q-1\).

Now consider the verifold \(Z^\prime \), with \(\partial Z^\prime = W\), \(Z^\prime = Z_{>1/2} \cup C_q\). We have

$$\begin{aligned} p{{\,\mathrm{-area}\,}}(Z^\prime ) \le \left( 1 - \left( \frac{1}{2}\right) ^p\right) p{{\,\mathrm{-area}\,}}(Z) + 2\pi \tau (p{{\,\mathrm{-area}\,}}(Z)) < (1+2\pi )p{{\,\mathrm{-area}\,}}(Z).\nonumber \\ \end{aligned}$$
(B.0.8)

Now define \(\overline{Z}\) to be the radially projected image of \(Z^\prime \) into \(S^{d-1}\). Since this projection has Lipschitz constant \(=2\), we obtain:

$$\begin{aligned} p{{\,\mathrm{-area}\,}}(\overline{Z}) \le 2^p(1+2\pi ) p{{\,\mathrm{-area}\,}}(Z) \end{aligned}$$
(B.0.9)

as desired. \(\square \)

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Freedman, M., Hastings, M. Building manifolds from quantum codes. Geom. Funct. Anal. 31, 855–894 (2021). https://doi.org/10.1007/s00039-021-00567-3

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