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Universality in long-distance geometry and quantum complexity

Adam R. Brown, Michael H. Freedman, Henry W. Lin, Leonard Susskind

arXiv:2111.12700v2hep-thmath.DGquant-ph

TL;DR

The paper asks whether geometries and quantum-complexity definitions that differ microscopically can become equivalent at long distances or high complexity. It studies homogeneous group-manifold metrics, develops low-dimensional examples and high-dimensional gate-complexity evidence, and concludes that broad universality classes are plausible, while quantitative equivalence and the critical schedule remain unsettled.

  • Problem

    The paper addresses whether short-distance metric choices can yield robustly equivalent long-distance geometry and quantum-complexity values beyond conventional polynomial equivalence.

    Source: Main conjectures · p. 4, Methods · p. 8

  • Method

    It analyzes homogeneous metrics through solvable low-dimensional examples and extends the comparison to high-dimensional gate and complexity geometries with varying penalty schedules.

    Source: High-dimensional gate example · p. 3, Discussion · p. 5

  • Results

    The study finds long-distance universality in low-dimensional Riemannian examples and presents evidence for analogous large-complexity universality in high-dimensional gate and complexity geometries.

    Source: Berger sphere · p. 2, Discussion · p. 5

  • Takeaways & Limitations

    Short-distance and long-distance geometry can decouple, supporting an effective-geometry program for classifying complexity metrics by their large-complexity behavior.

    Source: Discussion · p. 5

    The paper does not settle the quantitative closeness of the universality-class members or identify the critical schedule definitively.

    Source: Main conjectures · p. 4

Abstract

from arXiv · show

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds by their long-distance properties. We show that many metrics on low-dimensional Lie groups have markedly different short-distance properties but nearly identical distance functions at long distances, and provide evidence that this phenomenon is even more robust in high dimensions. An application of these ideas of particular interest to physics and computer science is complexity geometry--the study of quantum computational complexity using Riemannian geometry. We argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered. We conjecture that a new effective metric emerges at larger complexities that describes a broad class of complexity geometries, insensitive to various choices of microscopic penalty factors. We discuss the implications for recent conjectures in quantum gravity.

Motivating example

A tiny distance-preserving deformation of a homogeneous metric can radically alter short-distance volume while leaving long-distance distances nearly unchanged. This motivates long-distance universality as a decoupling between short- and long-distance geometry.

Motivating example

  • A deformation preserving homogeneity and changing every pairwise distance by at most one picometre can nevertheless make the volume infinite.The example separates a large change in volume from a tiny additive change in distances.Source: Motivating example · p. 2
  • The resulting short-distance geometry changes substantially, while the distance function at large separations remains insensitive to the deformation.The passage identifies this as a decoupling of short-distance and long-distance geometry.Source: Motivating example · p. 2

Low-dimensional Riemannian cases

Low-dimensional examples show that metrics with radically different short-distance structure can share nearly identical long-distance distances. The Berger sphere explains this through indirect paths and curvature, while the unicycle geometry exhibits the same universality across an unbounded distance range.

Berger sphere

  • Outside a shrinking near-origin region, Berger-sphere distances are largely independent of I, and the maximal separation remains exactly π for every I ≥ 1.The near-origin region ends near the cut locus, while indirect paths dominate farther away.Source: Berger sphere · p. 2
  • For Berger spheres with I = 1030 and I = ∞, distances agree to within a picometre despite the deformation sending the volume to infinity.The direct σz route becomes arbitrarily expensive, but cheap σx and σy motions can synthesize it indirectly.Source: Berger sphere · p. 2
  • As I grows, curvature hides increasing short-distance volume, although finite-resolution effective volume remains finite.The curvature length scales as I^-1/2, and the effective volume levels off once the probing scale is below that length.Source: Berger sphere · p. 2

Euclidean group

  • For the unicycle on SE(2), every metric with I ≥ 1 has the same large-z linear growth, with parking costs differing by at most O(1).The cost is unbounded because the curb distance z is unbounded, producing a genuine long-distance regime.Source: Euclidean group · pp. 2–3
  • At large I, unicycle parking has three regimes: steep linear growth, a square-root regime after the cut locus, and eventual linear growth with coefficient 1.The final regime arises because turning has a fixed cost while additional forward rolling contributes marginal cost one.Source: Euclidean group · p. 3
  • All I ≥ 1 metrics form a universality class whose leading long-distance behavior is I-independent despite radically different curvatures and cut loci.I acts as an irrelevant deformation, while I = 1 is a critical metric whose UV and IR growth agree exactly.Source: Euclidean group · p. 3
  • The unicycle example requires no endpoint tolerance because small directional changes provide enough wiggle room to correct the endpoint exactly.This contrasts with the perturbative estimate, which overestimates the cost of correcting errors.Source: Euclidean group · p. 3
  • The Berger sphere lacks a long-distance regime because its diameter is π, whereas the unicycle has an unbounded distance range and a flat critical geometry.The next examples are needed to obtain a non-trivial critical distance function.Source: Euclidean group · p. 3

High-dimensional gate example

The gate model exhibits a broad complexity universality class: sufficiently expensive penalty factors become irrelevant, while cheaper factors reduce complexity and can alter short-distance behavior without changing long-distance behavior.

High-dimensional gate example

  • A penalty factor becomes irrelevant once it exceeds the cost of synthesizing the same gate indirectly with cheaper gates.No individual gate is indispensable because gates of any fixed locality m ≥ 2 can reconstitute the others.Source: High-dimensional gate example · p. 3
  • The critical schedule is approximately set by dimension-counting bounds on the number of m-local gates needed to construct a k-local unitary.The lower bound is approximately saturated, up to overlap-induced U(2) isotropy overcounting by dimension 4.Source: High-dimensional gate example · p. 3
  • Making a penalty factor cheaper than its critical value reduces the complexity of almost all unitaries, whereas making it more expensive has little effect.Moderately cheaper factors can affect both short- and long-distance behavior, with the infrared effect summarized by a single multiplicative parameter.Source: High-dimensional gate example · pp. 3–4
  • All schedules no less expensive than the critical schedule approximately agree on complexities, forming a large universality class with the same large-distance behavior.The critical schedule is the unique member whose typical optimal compilation uses a single gate rather than a convoluted strategy.Source: High-dimensional gate example · p. 3
  • The critical schedule is conjectured to maintain linear complexity growth with one coefficient, while the cliff metric exhibits short-distance linear, intermediate sublinear, and long-distance linear regimes.For the cliff metric, the final linear regime has the critical metric’s lower slope.Source: Figure 2 · p. 4, High-dimensional gate example · p. 4
  • For the critical schedule, direct and indirect syntheses are approximately degenerate, making complexity robust against upward penalty deformations.This robustness follows from many equal-length compilation paths and load balancing across locality scales.Source: High-dimensional gate example · pp. 3–4

Application to complexity geometry

Complexity geometry is expected to exhibit stronger universality because its overcomplete path space makes many realizations of the same unitary available. The paper formulates two conjectures and proposes a three-regime quantitative picture distinguishing critical and cliff schedules.

Application to complexity geometry

  • Complexity geometry is more overcomplete than circuit compilation because paths may use superpositions of locality terms and change direction without penalty.This abundance of options is expected to make universality more robust than in the gate model.Source: Main conjectures · p. 4

Main conjectures

  • The first conjecture states that schedules no easier than a critical schedule may differ greatly in the UV but approximately agree on distance functions in the IR.The proposed universality class is anchored at the 2-local penalty normalization.Source: Main conjectures · p. 4
  • The critical schedule is conjectured to be the unique member whose UV and IR behaviors match, with typical geodesics remaining minimal for a time exponential in N.Its complexity growth therefore continues linearly with the same coefficient at short and long distances.Source: Main conjectures · p. 4
  • The paper does not settle how close schedules are quantitatively or uniquely identify the critical schedule, instead presenting an effective-geometry program and candidate answers.The critical and infinite-cliff schedules are treated as the easiest and hardest endpoints of the proposed class.Source: Main conjectures · p. 4

Quantitative conjecture for complexity growth

  • For a typical k-local Hamiltonian evolution, the critical schedule is conjectured to give direct geodesic complexity C = s = I_k z until z becomes exponentially large in N.The Hamiltonian is normalized so that z is the inner-product distance, and constant H_k generates a geodesic.Source: Quantitative conjecture for complexity… · p. 4
  • The infinite-cliff schedule is conjectured to show linear growth at tiny distances, sublinear growth after the cut locus, and renewed linear growth with the critical metric’s coefficient.The intermediate sublinear regime arises from indirectly synthesizing k-local operators through commuting 2-local Hamiltonians.Source: Quantitative conjecture for complexity… · pp. 4–5
  • A strong form of the conjecture bounds the cliff metric’s additive deviation from the critical complexity by the cost of one k-local gate.The paper also conjectures that the critical metric’s penalty factors grow at least exponentially with the number of qubits touched.Source: Quantitative conjecture for complexity… · p. 5

Discussion

The paper argues that short-distance geometry can decouple from an emergent long-distance metric, and extends this perspective from solvable low-dimensional examples to high-dimensional complexity geometries. It connects these conjectures to holographic complexity and proposes a broader program for classifying right-invariant geometries on large Lie groups.

Discussion

  • Short–long decoupling allows geometries with radically different local properties, including different volumes, to agree on distant-point distances within a picometre.The emergent long-distance metric depends on only a handful of relevant parameters rather than the full short-distance geometry.Source: Discussion · p. 5
  • The paper establishes long-distance universality in exactly solvable low-dimensional Riemannian examples and presents evidence for analogous universality in high-dimensional complexity geometries.The high-dimensional complexity-geometry claim is presented as a conjecture supported by the paper’s evidence.Source: Discussion · p. 5
  • If the complexity-geometry conjectures hold, many arbitrary penalty-factor choices become irrelevant in the semi-classical regime, potentially enabling a rigorous formulation of holographic complexity.The discussion links this possibility to holographic complexity correspondences18–21 and to Wilsonian renormalization ideas1.Source: Discussion · p. 5
  • The authors propose characterizing equivalence classes of right-invariant geometries on sufficiently large Lie groups, including non-compact groups and compact groups whose dimensions grow.This is identified as an interesting mathematical program beyond the paper’s complexity-geometry application.Source: Discussion · p. 5
  • The paper speculates that emergent geometry and long-distance universality may extend beyond unitary-state spaces to other rich high-dimensional systems, including many-body systems and deep neural networks.This broader relevance is presented as speculation rather than an established result.Source: Discussion · p. 5

Online content

The online material includes supplementary and reporting resources, while the cited literature spans renormalization, Riemannian geometry, quantum computation, complexity geometry, and holography. The article also provides publisher, licensing, and data-availability information.

Online content

  • Supplementary information, extended data, source data, reporting summaries, peer-review information, and data/code availability statements are provided online.The online-content notice directs readers to the article’s DOI for these materials.Source: Online content · p. 5

References

  • The cited references cover Wilson’s renormalization-group work on critical phenomena and the Kondo problem1, Ginzburg and Landau’s superconductivity theory8, Milnor’s Lie-group metrics2, and geometry-based quantum computation3, 4, 5, 6.Source: Online content · p. 5
  • The cited works address quantum control via geometry, single-qubit complexity geometry, Berger-sphere cut loci and diameter, exponential-operator decompositions, and sub-Riemannian geometry.Source: Online content · p. 5
  • The article discusses computational complexity and black-hole holography through cited works on complexity and black-hole horizons, shock-wave geometries, holographic complexity, bulk action, and action-based proposals.These topics appear in refs. 18–21.Source: Online content · p. 5

Publication information

  • The publisher’s note states neutrality on jurisdictional claims, and the article is distributed under a Creative Commons Attribution 4.0 International License.The licence permits reuse with attribution, subject to the stated conditions for third-party material.Source: Online content · p. 5

Methods

The methods develop a geometric framework for studying universality in complexity geometries and Lie-group metrics through right-invariant metrics, equivalence classes of distance functions, and large-scale curvature. They formulate conjectures and consistency checks for locality-dependent penalty schedules while connecting the framework to compilation bounds, holographic complexity, Wilsonian effective behavior, and coarse-geometric equivalence.

Review of complexity geometry

  • Complexity geometry assigns U(2^N) a smooth right-invariant distance whose diagonal penalty factors depend only on generalized-Pauli operator weight k, with analogous constructions for Majorana fermions.The metric stretches difficult directions according to a penalty schedule I_k, while k-local Hamiltonians may act across all qubits through weight-k terms.Source: Methods · p. 6

The critical metric

  • The proposed critical schedule has steadily increasing locality penalties with an exponentially large maximum, making an exponential metric a candidate approximation up to sub-exponential corrections.The exponential metric has sectional curvature magnitude below O(1), unlike the cliff metric’s huge curvature from easy 2-local directions commuting into hard 3-local directions.Source: Methods · p. 6
  • The conjectures pass diameter checks: infinite-cliff and exponential metrics have exponentially large diameter bounds, supporting approximate agreement at long distances but not yet complexity agreement for almost all complex unitaries.The paper explicitly notes that diameter agreement is weaker than showing comparable complexity for almost all sufficiently complex unitaries.Source: Methods · pp. 6–7
  • A heuristic 2-local compilation strategy yields linear-in-k cost for monomial directions and argues that the binomial metric lies in the infinite-cliff universality class, while the key error-correction step remains heuristic.The strategy exploits generalized-Pauli operators generated recursively by nested commutators; a weaker provable result supports approximation but not the conjectured exact linear bound.Source: Methods · p. 7

Relation to black holes and holography

  • If the conjectures hold, long-distance universality would make holographic complexity robust against many penalty-factor choices in the semiclassical limit, potentially enabling a rigorous formulation of holographic complexity.This addresses the conventional concern that numerical complexity depends sensitively on arbitrary primitive-gate or penalty-factor definitions.Source: Methods · p. 8

Wilsonian connections and coarse geometry

  • The paper interprets penalty schedules as Wilsonian couplings and defines coarse equivalence for metric sequences using bounds with constants independent of system size, while noting that the analogy is suggestive rather than exact.Penalty factors above their critical values are irrelevant to infrared behavior; for bounded compact groups, coarse equivalence becomes meaningful only with uniform bounds across the sequence.Source: Methods · pp. 8–9

Methods

  • Different generating sets or penalty factors can define distance functions in the same coarse equivalence class, exemplified by Cayley-graph geometry [43].The framework treats δ-hyperbolicity as a large-scale negative-curvature property, with free groups providing a tree-like example [45].Source: Methods · p. 9
  • The authors propose extending geometric group theory to non-compact Lie groups and sequences of compact Lie groups, where few coarse-geometry classes may represent infinitely many right-invariant metrics.They expect large-scale curvatures across an equivalence class to approximately agree with those of a critical metric, despite extreme local curvatures; this supports the conjecture of negative sectional curvature [7, 38].Source: Methods · p. 9

1 Zooming in and zooming out

The paper frames zooming out as a way to classify metrics by coarse, large-scale geometry rather than short-distance details. Low-dimensional calculations suggest rapid convergence and motivate conjectures about broad universality phases, critical metrics, and renormalization flows for complexity geometry.

Evidence for large-scale universality

  • Low-dimensional calculations converge faster than proven estimates and suggest broad metric phases with nearly constant large-scale geometry separated by sharp transitions.The authors emphasize that the scope of the additive estimates remains unknown, especially for U(2^N).Source: §1 · p. 12

Coarse-geometric analogy

  • Large-scale behavior in related geometries is controlled, to a tolerance between additive and multiplicative error, by the metric on the Lie-algebra Abelianization.The authors propose that bracket-complete easy directions in u(2^N) play an analogous role for complexity geometry.Source: §1 · p. 13

Sub-Riemannian reduction

  • The proposed complexity-geometric program retains an easy subspace E, forbids motion outside it to define g0, and seeks criteria ensuring efficient bracket generation.A conjectural inequality is proposed when E is effectively bracket complete, with C_g0 of order O(1) in that regime.Source: §1 · p. 14

Conjectural comparison

  • For large scales, the authors conjecture that a sub-Riemannian metric can remain additively close to a Riemannian metric agreeing on E, but they lack proof tools.The conjecture assumes sufficiently large penalties in orthogonal directions and faces fundamental obstacles.Source: §1 · p. 14

Critical metric

  • For two-qubit-easy geometries, the conjectural critical schedule lies at the lower edge of a phase whose upper limit is the sub-Riemannian metric.Increasing penalties for k≥3 should leave already large distances nearly unchanged, whereas reducing some penalties can shrink them at first order.Source: §1 · p. 15

Renormalization perspective

  • The proposed renormalization flow lowers higher-weight penalties while rescaling to preserve diameter, potentially converging toward a critical metric.The authors distinguish this speculative flow from Ricci flow because the conjectured fixed points are not Einstein metrics.Source: §1 · pp. 16–17

Obstacles and open questions

  • Abnormal geodesics and unresolved high-frequency discrepancies limit the conjecture that long-distance metrics remain smoothly close at diameter-scale separations.The authors know of no analogous abnormality on U(2^N), but explicitly have no proof that rich geodesic dynamics exclude it.Source: §1 · pp. 15, 17

Broader program

  • The program aims beyond complexity geometry toward coarse classification of metrics on non-compact Lie groups and sequences of compact Lie groups.This would generalize existing coarse-geometry work on finitely generated discrete groups.Source: §1 · p. 17

2 Holography, load balancing, and the critical metric

The paper relates its critical-metric conjecture to holographic complexity and examines whether particular penalty schedules realize that role. It also highlights approximate load balancing as a possible bridge to bulk physics while separating penalty-factor dependence from quantum-field-theory regulator dependence.

Critical metric and holography

  • The exponential metric is favored over the binomial metric as a critical-metric candidate because its penalties depend only on locality k, not system size N.The binomial candidate would imply non-extensive complexification rates for k-local Hamiltonians across all k, unlike Brownian circuits, gates, and black holes.Source: §2 · p. 18

Load balancing

  • The conjectured critical metric is approximately load-balanced, with many paths of roughly equal complexity connecting distant states.The paper identifies the possible meaning of this degeneracy for bulk physics as an open direction.Source: §2 · p. 18

Scope distinction

  • Penalty-factor sensitivity in complexity should not be conflated with sensitivity to the ultraviolet regulator of the underlying quantum field theory.The paper notes that even UV-finite models such as SYK still require a choice of complexity penalty factors.Source: §2 · p. 19

3 Two-dimensional example

The two-dimensional example uses a deformed Bloch sphere to visualize how microscopic metric changes can strongly affect volume while barely changing long-distance state complexity. Increasing the penalty for vertical-axis rotations makes equatorial motion expensive and drives the volume logarithmically to infinity.

Construction

  • The deformed Bloch-sphere metric models single-qubit state complexity as the least unitary complexity connecting two states.Its line element is specified by Eq. S.3, with deformation parameter I controlling the penalty structure.Source: §3 · p. 19

Penalty deformation

  • For I>1, rotations with a vertical σ_z component become more expensive, while equatorial rotation axes remain comparatively accessible.As I grows, only rotations with nearly equatorial axes avoid prohibitively large costs.Source: §3 · p. 19

Long-distance universality

  • As I→∞, distances change only by a tiny additive amount while the volume diverges logarithmically.The divergence arises because neighboring equatorial points acquire a very large distance factor.Source: §3 · pp. 19–20
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