Switchbacks and the Bridge to Nowhere

1 Introduction

• One-sided black holes can possess growing ERBs whose volume tracks the increasing computational complexity of the quantum state.This extends the correspondence previously discussed for two-sided black holes.Source: §1 · p. 2
• The paper addresses the apparent tension between a pure, unentangled black hole and an ERB by introducing a growing “bridge-to-nowhere.”The proposed one-sided bridge grows with time while tracking complexity.Source: §1 · p. 3
• Because black holes evolve through Hamiltonian dynamics rather than random circuits, gate-counting complexity is presented as an imperfect model for their evolution.The paper therefore seeks a complexity description better suited to Hamiltonian systems.Source: §1 · p. 3
• Nielsen and collaborators’ continuum complexity geometry [5][6] is proposed as a tool for analyzing black-hole evolution.The paper applies this framework to fast-scrambling and the switchback effect associated with precursors and shockwave geometries [1] [9].Source: §1 · p. 3
• The methods apply to both one- and two-sided cases, although the paper focuses on a single gauge-theory copy and a one-sided black hole.This is the stated convention for definiteness rather than a restriction of the methods.Source: §1 · p. 3

2 The Bridge to Nowhere

• The construction distinguishes one- and two-sided ERBs: entanglement can stitch two one-sided ERBs together, whereas two unentangled black holes have no internal connection.The paper uses ERB1 and ERB2 for the one- and two-sided cases.Source: §2 · pp. 3, 5, 6
• The one-sided ERB is defined as the maximal-volume spacelike hypersurface ending on a chosen boundary sphere at time t.This construction parallels the two-sided case [1].Source: §2 · pp. 3–4
• The maximal-surface equations are the same as for the two-sided black hole [1], and the resulting one-sided behavior is similar.The spatial volume requires regulation because it diverges at the ADS boundary; one option is counting only volume behind the horizon.Source: §2 · p. 4
• At late times ERB1 approaches a fixed hypersurface, while finite-time growth occurs through stretching just inside the horizon.The green curve in Figure 2 represents the asymptotic late-time limit.Source: §2 · p. 4, Figure 2 · p. 5
• After the shell forms the black hole, the ERB1 volume grows linearly with time, paralleling the linear growth of computational complexity.The complexity growth rate is stated as the black-hole entropy S times temperature T.Source: §2 · p. 6
• Before shell crossing, ERB1 is a constant-time ADS slice; after crossing the shell, the portion beyond it is modified.The shell process is depicted in the Penrose diagram of Figure 1.Source: §2 · pp. 4, 6, Figure 1 · p. 4

3.1 Operator Complexity

• Operator complexity is modeled as shortest-path length in SU(2^K), with a cost metric that makes three-or-more-qubit directions expensive.Nielsen’s continuum construction replaces discrete gates with Hamiltonian-generated paths and geodesic distance.Source: §3.1 · pp. 9–11
• The C-geometry is strongly anisotropic: geodesics avoid hard directions, and for q > 4K complexity becomes insensitive to the precise cost parameter.Most sectional curvatures are negative, so neighboring geodesics tend to separate exponentially.Source: §3.1 · pp. 11–12
• For a fixed easy Hamiltonian, complexity grows linearly with time until a critical time s_c, when the original geodesic ceases to be shortest.This establishes rather than merely assumes early-time linear growth.Source: §3.1 · p. 11

3.2 State Complexity

• State complexity is defined through the least-complex unitary mapping a simple reference state to the target, equivalently as minimum geodesic distance on projective Hilbert space.Different unentangled reference states remain within distance K, corresponding to one parallel-computing time step.Source: §3.2 · pp. 12–13
• Quantum states typically have complexity of order e^K, require exponential time to reach, and take doubly exponential time to return to simplicity.This contrasts with classical configurations, whose maximum complexity is K/2 and whose recurrence time is 2^K.Source: §3.2 · pp. 13–14
• For an AdS black hole, the ERB volume is proportional to state complexity, whose growth rate is tied to the entropy and temperature through S and T.The relation is reported as confirmed for the thermofield-double state and many shockwave deformations.Source: §3.2 · pp. 14–15

4.1 Size and Complexity of Precursors

• Precursors probe the complexity–ERB connection, and the section compares ERB response, random circuits, and C-geometry as three approaches to precursor complexity [1][5][6][9].Precursor size measures how strongly other qubits are causally influenced by W, while complexity is denoted CW(t).Source: §4.1 · p. 15

4.2 Quantum Circuit Analysis

• In the Hayden–Preskill circuit, precursor size grows exponentially while small, saturates near K at the scrambling time, and precursor complexity sums the epidemic sizes.The scrambling time is t∗= l log K, and the size approaches its asymptotic value s →K.Source: §4.2 · pp. 17–19

4.3 Switchback Effect

• The switchback effect removes gates that cancel between U(t) and U(−t), matching the cancellation found in Shenker–Stanford shockwave calculations [1].The same effect is later reproduced by Nielsen’s geometric treatment of complexity.Source: §4.3 · pp. 19–20

4.4 Complexity-geometry of Precursors

• A precursor traces vertical evolution geodesics joined by a short horizontal W segment, while varying connecting geodesics generates a two-dimensional C2 surface.The horizontal geodesic lengths represent precursor complexity; the proposed metric is difficult to derive directly and is instead inferred and tested.Source: §4.4 · p. 21

4.5 Complexity Calculation

• Uniform negative curvature in C2-geometry reproduces the quantum-circuit precursor complexity, including the switchback shift caused by minimum geodesic distance [5][6].The geometric result is described as significant evidence that precursor complexity is well captured by a C2-geometry with uniform negative curvature.Source: §4.5 · pp. 23–24

4.6 Size of Precursor

• The proposed size geometry represents precursor size through tangent-line distances or approximately equivalent areas, but its geometric interpretation remains tentative.The area ratio is close to unity for times below scrambling, motivating speculation that an appropriately defined S is represented by the area A.Source: §4.6 · pp. 25–27
• The three precursor-size functions are compared using plots of their values and ratios.Figure 10 plots the functions, while Figure 11 plots F2/F1 and F3/F1.Source: §4.6 · p. 28
• Functions 4.43 and 4.41 are extremely close over most of the range, whereas 4.42 differs substantially.The same comparison is visible in the ratios F2/F1 and F3/F1.Source: §4.6 · p. 28
• It remains unclear why F3 provides a better fit to F1 than F2.The paper explicitly leaves the explanation for this fitting difference unresolved.Source: §4.6 · p. 28

• Hawking radiation should turn the bridge-to-nowhere into a bridge-to-somewhere purified by the radiation, whose properties and circuit description remain open.Source: §5 · p. 29
• An easy-Hamiltonian trajectory is geodesic but not always shortest; cut points may appear by the classical recurrence time and signal breakdown of the classical ERB description.Source: §5 · p. 29
• The proposed C2 geometry captures expected precursor properties, but lacks a first-principles derivation and a demonstrated curvature of order −1/K^2.Source: §5 · pp. 29–30
• The analysis is restricted to black holes whose radius equals the AdS radius l, leaving its extension to larger black holes partly unresolved.Source: §5 · p. 30
• The broader unresolved issue is how much of general relativity is encoded in quantum complexity beyond the parallels already seen for ERB growth, precursors, and switchbacks.Source: §5 · p. 30

• The appendix calculates the functional form of ERB1 for the one-sided version of a BTZ black hole.Source: A Geometry of ERB1 · p. 30

• The one-sided AdS black hole is formed by gravitational collapse of a spherical shell of null matter, with empty AdS before and BTZ after the shock wave.Source: §A.1 · pp. 30–31
• The radial coordinate remains continuous across the shock wave, while boundary time coordinates are chosen continuous at r = ∞.Source: §A.1 · p. 31
• Null coordinates are introduced separately in the AdS and BTZ regions so the matching condition requires continuity of both r and v across the shock wave.Source: §A.1 · pp. 31–32

A.2 Extremal Surface

• The extremal surface ends at boundary time t_b and is a constant-time disk in pure AdS when t_b < 0.Source: §A.2 · p. 32
• The post-shock surface consists of an AdS disk and a BTZ segment parameterized by r = r(v), with conserved energy because the Lagrangian is independent of v.Source: §A.2 · p. 32
• Across the shock wave, extremality requires continuity of the radial coordinate and its conjugate momentum p_r.Source: §A.2 · p. 33
• At large time, the limiting surface lies entirely inside the horizon and approaches the r = r_m surface exponentially in the boundary-time coordinate.Source: §A.2 · pp. 34–35
• The large-time area grows linearly because the surface spends most of its time near r = r_m, while the outer region contributes an approximately constant area.Source: §A.2 · pp. 35–36