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Quantum complexity in gravity, quantum field theory, and quantum information science
Stefano Baiguera, Vijay Balasubramanian, Pawel Caputa, Shira Chapman, Jonas Haferkamp, Michal P. Heller, Nicole Yunger Halpern
TL;DR
The paper addresses how quantum complexity can characterize long-time quantum evolution and relate to quantum-information quantities, geometric dynamics, and holographic observables. It surveys circuit, geometric, spreading-based, and holographic approaches, emphasizing results on complexity growth and the constraints on establishing exponential-time growth.
Problem
Quantum complexity must be related across quantum circuits, unitary geometry, dynamical spreading, and holographic observables, despite tensions between growing gravitational quantities and quickly equilibrating local observables.
Method
The paper synthesizes circuit-complexity results, geodesic methods on unitary groups, state-spreading constructions, and holographic complexity proposals.
Results
The review reports linear growth and eventual saturation for exact random-circuit complexity, finite-distance conjugate points for local-Hamiltonian geodesics, and model-dependent complexity-growth bounds in SYK systems.
Takeaways & Limitations
Complexity growth depends on the dynamics and geometric structure: shortcuts truncate growth in simple systems, while chaotic SYK dynamics can support growth for exponential times.
Takeaways & Limitations
Proving superpolynomial complexity growth for exponentially long local-Hamiltonian evolution appears to require establishing the unproved separation PSPACE ⊄ BQP/poly.
Abstract
from arXiv · showhide
Quantum complexity quantifies the difficulty of preparing a state or implementing a unitary transformation with limited resources. Applications range from quantum computation to condensed matter physics and quantum gravity. We seek to bridge the approaches of these fields, which define and study complexity using different frameworks and tools. We describe several definitions of complexity, along with their key properties. In quantum information theory, we focus on complexity growth in random quantum circuits. In quantum many-body systems and quantum field theory (QFT), we discuss a geometric definition of complexity in terms of geodesics on the unitary group. In dynamical systems, we explore a definition of complexity in terms of state or operator spreading, as well as concepts from tensor-networks. We also outline applications to simple quantum systems, quantum many-body models, and QFTs including conformal field theories (CFTs). Finally, we explain the proposed relationship between complexity and gravitational observables within the holographic anti-de Sitter (AdS)/CFT correspondence.
1. Preface
This review integrates quantum complexity across quantum information, QFT, many-body physics, and gravity, where different fields use distinct definitions and analytical tools. It organizes these perspectives around circuit, geometric, spreading, and holographic approaches, while highlighting proposed links between complexity and gravitational observables.
- The review aims to build bridges among gravity, quantum information theory, QFT, and quantum many-body physics through quantum complexity.These fields study dynamical phenomena with different notions of complexity, methods, and tools.
- Quantum complexity originated as a resource measure for the time, space, or gate requirements of quantum algorithms.The review traces this framework from quantum computation toward broader physical applications.
- Holographic complexity extends the AdS/CFT discussion beyond entanglement because some gravitational geometries continue evolving after entanglement-related quantities equilibrate.Black-hole spatial volumes can grow to times exponential in the black-hole entropy.
- Review structure: The review follows paths covering circuit complexity, continuous-time geometric complexity, state and operator spreading, and applications to quantum information and many-body physics.Its structure begins with a glossary and proceeds through dialogue, information-theoretic, geometric, dynamical, and holographic material.
- Holographic connections: Proposed gravitational duals reproduce features associated with evolving complexity, including late-time linear growth, switchback delays, and exponentially long saturation in selected models.The review also notes a precise relation between spread complexity and wormhole length in two-dimensional gravity.
4. Definition and time evolution of quantum complexity
Quantum complexity measures the resources needed to prepare states or implement unitaries, and its time evolution can be analyzed through circuit counting, scrambling, and unitary-design arguments. The review describes linear growth, saturation, switchback behavior, and limits on proving late-time lower bounds.
- Time evolution: Complexity grows linearly for a time exponential in system size, saturates at an approximately constant value until a doubly exponential time, and later undergoes Poincaré recurrences.
- Time evolution: A small perturbation produces a switchback effect by delaying linear complexity growth by the scrambling time.
- Circuit counting: For fast-scrambling 2-local Hamiltonians, discretized evolution is believed to provide an optimal circuit up to exponentially large times because cancellations are unlikely.
- Equilibration and limitations: The review notes that complexity saturation is a late equilibration stage, while terminology differs across authors and the illustrative list is not comprehensive.
- Definition: Quantum complexity is the least number of quantum gates needed to prepare a target state or implement a target unitary.
- Random circuits: Random-circuit counting and unitary-design arguments establish linear exact-complexity growth up to exponentially deep circuits and approximate-complexity lower bounds of Ω(Kt) with high probability.
5. Paradigms for complexity I: Nielsen complexity
Nielsen complexity models the difficulty of implementing unitaries as shortest paths on a metric-equipped unitary group, with metrics assigning higher cost to complex operations. The framework also induces state complexity by quotienting out transformations that leave a state unchanged up to global phase.
- Complexity geometry: The complexity metric assigns large norms to complex operations and small norms to simple ones, encoding implementation difficulty geometrically.Different metric choices parallel different gate-set choices and can share the same long-distance behavior.
- Nielsen complexity: Nielsen complexity is the shortest-geodesic length from the identity to a target unitary under a chosen complexity metric.The metric assigns costs to infinitesimal operations, and the minimum is taken over all trajectories connecting the identity and target unitary.
- Relation to circuit complexity: Nielsen complexity upper-bounds approximate circuit complexity and lower-bounds exact circuit complexity.This connects the geometric quantity to discrete circuit notions while retaining a continuous formulation related to Trotterized evolution.
- Solvable settings: Nielsen complexity can be exactly or approximately solvable for unitary representations of symmetry groups relevant to free systems and conformal field theories.Examples include symplectic, orthogonal, global conformal, and Virasoro-related constructions.
- Optimization over trajectories: The framework applies to physical or auxiliary-time trajectories generated by Hamiltonians, whose tangent components act as control functions on the unitary manifold.Different trajectories represent different ways of generating the same target unitary, and the optimal one minimizes integrated cost.
- State complexity: State complexity is constructed from unitary complexity by minimizing the cost of producing a target state from a fixed reference state.Because global phases are physically irrelevant, transformations in the state stabilizer are assigned zero cost, yielding a quotient related to complex projective space.
Equivalence classes of metrics and the universality of penalty schedules.
The review relates complexity growth to geodesic structure on the unitary group, showing how penalty schedules and operator properties constrain when linear growth can persist or terminate. It also connects binding complexity to entanglement through relations that can become independent of penalty factors.
- Complexity growth: Linear Nielsen-complexity growth can persist for exponential times when conjugate points are pushed to infinity or when the Eigenstate Complexity Hypothesis holds.The cited criterion gives persistence for O(e^ϵS), with S the logarithm of the Hilbert-space dimension.
- Geodesic minimality: Shortest-geodesic analysis is difficult because multiple geodesics can connect the same unitary, and a later path may be shorter due to group topology.In SU(2), the linear trajectory eventually reaches the opposite pole and complexity decreases along a shorter route; analogous loop obstructions are expected after exponential times in larger groups.
- Complexity growth: For arbitrary local Hamiltonians with finite penalty, conjugate points must occur at finite distance along the linear geodesic.The result follows from persistence under increasing penalty together with Morse theory on the path space.
- Complexity growth: Adjoint eigenoperators in easy and hard directions generate conjugate points at times 2πZ/λ and 2π(1+µ)Z/λ′, respectively.The hard-direction result includes the penalty factor µ, whereas the easy-direction result does not.
- Binding complexity: Binding complexity can become independent of nonlocal-generator penalties and obey a universal relation with subsystem entanglement.For p=1 under specified generator assumptions, the resulting fixed function of Schmidt coefficients also supports bounds involving entanglement entropy.
6. Paradigms for complexity II: Krylov and spread complexities
Krylov and spread complexities quantify how operator or state dynamics explore Hilbert space rather than only how many operations implement a process. The review develops their computational constructions, including Lanczos methods, and relates them to chaos, integrability, and holography.
- Conceptual framework: Krylov complexity measures operator growth, while spread complexity measures how an initial state explores Hilbert space during unitary evolution.The two notions are related but suited to different questions about dynamical spreading.
- Analytical tools and applications: The review presents numerical methods, symmetry-based analytical results, and formulas connecting density of states with Lanczos coefficients.It also discusses random-matrix correlations in Lanczos coefficients and applications to chaos and integrability.
- Holographic connection: The review identifies a precise relation between JT-gravity wormhole length and the spread complexity of the dual SYK thermofield-double state.This connection appears in the holography discussion alongside the broader treatment of Krylov and spread complexities.
- Conceptual framework: Spread complexity focuses on the support of a single evolving state in a chosen basis rather than the spreading of nearby states measured by OTOCs.This reframes dynamical exploration as the breadth of one trajectory through Hilbert space.
- Krylov construction: The Krylov basis minimizes the number of basis elements supporting a discretely time-evolving state.Each new basis vector is chosen from the component of the evolved state orthogonal to the previously constructed span.
- Krylov construction: Lanczos orthonormalization constructs the Krylov basis from {H^k|Ψ0⟩}, and its termination dimension measures the subspace explored by the evolving state.The Hamiltonian acts tridiagonally in this basis, with b_K=0 marking termination.
Tridiagonalization.
The review develops tridiagonalization-based complexity tools for relating density of states, Lanczos coefficients, operator or state spreading, and diagnostics of chaos across physical systems. It also describes geometric mappings, random-matrix results, and limits on interpreting complexity growth as a chaos signature.
- Universal analytical formulas relate the density of states to coarse-grained Lanczos coefficients, with applications to SYK and random matrix ensembles.The formulas can also be inverted to recover the Lanczos spectrum from density-of-states data.
- Spectral correlations in chaotic random-matrix theories produce a complexity peak and subsequent decline, whereas matching the density of states without eigenvalue correlations yields plateauing without that peak.The peak and slope are associated with spectral rigidity.
- Krylov dynamics map quantum evolution onto effective one-dimensional hopping chains, while Fubini–Study geometry maps selected operator growth to geodesic motion.For SL(2,R), SU(2), and Heisenberg-Weyl systems, Krylov complexity is proportional to the enclosed phase-space volume.
- In generic finite-dimensional non-integrable systems, Krylov complexity grows exponentially, then linearly after t ≥ log(K), and finally saturates at exponentially large times t ∼ e^K.These regimes correspond respectively to growing, saturated, and decreasing Lanczos coefficients.
- For generic initial operators or states, the late-time plateau cannot generally distinguish chaos from integrability because it is determined by universal density-of-states information.Lanczos-coefficient fluctuations and covariances can still distinguish the regimes even when plateau values coincide.
- At large q, the Krylov exponent equals the OTOC Lyapunov exponent at leading order and both approach 2π/β at low temperatures, but equality fails at order 1/q.Flow SYK examples also show transitions that the OTOC exponent detects but the Krylov exponent may not.
Krylov complexity in QFT.
In continuum QFT, Krylov-complexity diagnostics face complications because exact correlators can produce operator-growth patterns that resemble nonchaotic behavior. Appropriate UV and IR cutoffs may clarify this behavior, but distinguishing chaos from integrability remains unresolved.
- In 2D CFTs, thermal two-point correlators are identical in integrable and chaotic theories, while their Lanczos coefficients grow linearly at large index.This provides a counterexample to a universal operator-growth expectation based on these correlators.
- UV cutoffs may isolate chaotic or scrambling behavior in Lanczos coefficients above the cutoff scale Λ.
- IR cutoffs can separate even and odd Lanczos coefficients, producing linearly growing sequences with different intercepts.
- Distinguishing chaotic from integrable operator growth in continuum QFT remains an open problem, with proposed routes including von Neumann algebras, Lanczos-coefficient covariances, and microcanonical projections.A direct connection between algebraic-QFT methods and Krylov-basis methods has not been established.
- The relation between Krylov and Nielsen complexity remains debated because Krylov complexity can correspond to a Fubini–Study volume rather than a distance.
Connecting Nielsen and Krylov complexities.
Extending spread and Krylov complexity to time-dependent Hamiltonians is identified as an important direction for studying driven systems and external sources.
- Extending spread and Krylov complexity to time-dependent Hamiltonians is important for understanding driven systems.
Other systems.
The review connects complexity to open-system dynamics and holographic gravity. It highlights how external baths affect entanglement spreading and how spread complexity can relate to geometric quantities in gravity.
- Open quantum systems are governed by Lindbladian evolution of the density matrix when external sources or baths are included.
- Interaction with an external bath affects the structure of entanglement spreading in open quantum systems.
- In double-scaled SYK, spread complexity is directly related to the length of dual wormholes in two-dimensional Jackiw–Teitelboim gravity.
- The review motivates holographic complexity by considering black-hole microstates that may be hidden because simple probes cannot discriminate between them.
7. Quantum complexity and space-time: a more concerted approach
The review compares holographic complexity proposals with quantum-complexity dynamics, emphasizing geometric constructions that reproduce characteristic growth and scrambling behavior while retaining ambiguities and open questions about precise duality.
- Open questions: The proposals remain qualitatively matched to complexity dynamics, but a precise equation identifying a particular gravitational observable with a CFT complexity remains an open question.The geometric quantities are diffeomorphism-invariant, yet the review asks whether they map to a specific complexity notion beyond shared qualitative behavior.
- Holographic complexity proposals: Holographic complexity proposals use extremal geometric objects, including maximal-volume surfaces, WDW patches, and regions selected by boundary extremization.CV uses a maximal-volume codimension-one surface; CA and CV2.0 use the WDW patch; CAny extremizes a bulk region with codimension-one boundaries.
- Holographic complexity proposals: The CV, CA, and CV2.0 proposals contain scale or normalization ambiguities, while CA and CV2.0 also probe poorly understood regions near black-hole singularities.CV and CV2.0 require ℓbulk, CA involves null-boundary normalization and counterterm scales, and semiclassical validity near singularities is uncertain.
- Holographic complexity proposals: CAny selects a bulk region by extremization and then computes a gravitational observable, defining a broad class compatible with linear growth and the switchback effect.The conjecture uses scalar functions to govern the extremization and observable construction.
- Time-dependent properties: All reviewed holographic proposals reproduce late-time linear growth and the switchback effect in black-hole backgrounds.These behaviors match those found for circuit, Nielsen, and Krylov/spread complexity.
- Time-dependent properties: The switchback delay is controlled by the scrambling time t∗ = 1/(2πT) log(M/E) ≈ 1/(2πT) log S for a low-energy shock.At late times, complexity grows linearly with tL + tR after a scrambling-time delay.
Behavior at early and intermediate times.
Shock-wave studies reveal an early-time plateau followed by late-time linear growth, with the plateau length encoding scrambling and depending on perturbation energy.
- Early and intermediate times: At early times tw ≪ t∗, CV and CA exhibit an approximately constant complexity plateau, while at later times tw ≫ t∗ they approach linear growth.The plateau becomes longer for lower-energy perturbations.
- Early and intermediate times: The plateau length provides an estimate t∗ ≈ 1/(2πT) log(M/T) of the scrambling time.Here T is the Hawking temperature.
- Comparison with circuit models: The holographic observables' behavior agrees with the circuit-model result and extends earlier studies to shocks with energies above the thermal scale.The review identifies the matching by comparing the corresponding figures.
- Figure 11: Figure 11 plots ΔCV − ΔCV,NS against the shock insertion time −tw, with colors encoding increasing shock energy.The displayed energy ratios range from 10−8 to 1.5.
- Scope: The analysis is semiclassical and therefore explores holographic complexity only up to times exponential in the black-hole entropy.Quantum-gravity corrections are not included in the black-hole geometry used for the derivation.
Beyond semiclassical gravity: late-time behavior.
Beyond semiclassical gravity, quantum-complexity proposals are connected to nonperturbative geometric observables and tested across subregions, defects, boundaries, and de Sitter settings. These studies reveal late-time growth and saturation patterns, ultraviolet divergences, and limits on identifying complexity with entanglement entropy.
- Beyond semiclassical gravity: late-time behavior.: The regulated time-dependent ERB length is finite and regulator-independent, grows linearly, and saturates at times and values of order e^S0 in JT gravity.Its variance remains negligible near t ∼ O(e^S0) but reaches the same order as its mean near t ∼ O(e^2S0).
- Beyond semiclassical gravity: late-time behavior.: The spectral decomposition of ⟨e^−∆ℓ⟩ acts as a generating function for quantum complexity and yields linear growth followed by late-time saturation as ∆→0.Its generating function has a slope-ramp-plateau structure, with the ramp disappearing in the ∆→0 limit.
- Time-independent properties of holographic proposals.: UV divergences are robust across holographic and free-field descriptions, with complexity divergences proportional to system volume and subleading terms encoding boundary-slice geometry.The subleading divergences occur in jumps of two powers of the cutoff and agree with free-QFT results.
- Subregions.: For subregions, CV complexity is computed from maximal surfaces bounded by the region and its HRT surface, while the leading divergence scales as kdCTV(B)/δd−1.Subregion proposals are generally superadditive in pure states; CA can be tuned between subadditivity and superadditivity through Lct, but positivity of its leading divergence makes it superadditive.
- Subregions.: A simple linear relation between complexity and entanglement entropy fails for multiple BTZ segments and is not expected in dynamical situations because the quantities evolve differently.The multiple-segment dependence also varies intricately with subregion sizes and temperature.
- Defects and boundaries.: Defects and boundaries produce logarithmic complexity-of-formation divergences in 2+1 dimensions, while Janus geometries show scheme-independent logarithmic or finite terms depending on dimension.For Janus AdS, d = 2, 4 yield logarithmic divergences, whereas odd dimensions yield a scheme-independent finite term.
Holographic complexity in de Sitter space.
In de Sitter holography, complexity proposals generally exhibit unusual late-time behavior, including finite-time divergences, while Krylov-based complexity connects more quantitatively to bulk gravitational observables in lower-dimensional models. These correspondences remain limited by dimensionality and setup-dependent definitions.
- Holographic complexity in de Sitter space: CV, CV2.0, and CA complexity proposals in de Sitter space can diverge at a finite critical time, interpreted as hyperfast complexity growth.The interpretation associates this behavior with circuits whose gates involve many qubits at each time step.
- Holographic complexity in de Sitter space: The CV observable in empty de Sitter space depends on how the relevant surface is anchored and can cease to exist at a regulator-dependent critical time.Anchoring to a stretched horizon gives tcrit proportional to L arctanh(rst/L), and the critical time can become arbitrarily large near the de Sitter horizon.
- Holographic complexity in de Sitter space: Finite-energy shock waves in asymptotically de Sitter black holes produce a complexity plateau around t = 0 whose duration increases when the shock crosses the stretched horizon earlier.This behavior is analogous to the switchback effect discussed in AdS.
- Krylov and spread complexity: In double-scaled SYK, spread complexity grows linearly forever in the double-scaling limit, while finite-dimensional Lanczos coefficients eventually decline and force late-time saturation.The decline is interpreted as a non-perturbative quantum-gravity correction to the early-time classical growth.
- Krylov and spread complexity: Spread complexity matches a bulk volume across the full DSSYK regime only after including Euclidean state preparation and quantum corrections to the gravitational proposal.The correction is required for quantitative agreement with boundary spread complexity.
- Krylov and spread complexity: Krylov-basis complexity can correspond to bulk radial momentum only in proper radial distance coordinates, while higher-dimensional generalization remains an open next step.The existing comparisons rely largely on low-dimensional toy models, including AdS2 and double-scaled SYK.
8. Paradigms for complexity III: tensor-network-inspired definitions of complexity
Path-integral optimization extends tensor-network intuitions to QFT by treating the background metric as a gate-density profile and minimizing an associated Liouville action. In holographic settings, this optimization matches geometric constructions in AdS and admits a gravitational action completion.
- 8.1. Complexity from path-integral optimization: Path-integral complexity uses Euclidean path integrals and tensor-network-inspired geometry to define complexity for QFT wavefunctions.The approach was motivated by optimized tensor networks such as MERA and cMERA.
- 8.1. Complexity from path-integral optimization: The background metric is interpreted as an unoptimized density of continuous-tensor-network gates, parametrized in two dimensions by a Weyl factor ϕ(τ, x).The boundary condition restores the original flat metric at the regulated time slice.
- 8.1. Complexity from path-integral optimization: The optimal path integral minimizes the Liouville action with respect to the Weyl factor, equivalently selecting metrics that solve the Liouville equation.The proportionality factor between wavefunctions is expressed through the Liouville action, with the central charge entering the construction.
- 8.1. Complexity from path-integral optimization: For vacuum and thermal CFT states, path-integral optimization produces hyperbolic geometries, including the Poincaré plane, Poincaré disc, and hyperbolic strip.Higher-dimensional optimization over the Weyl factor likewise yields metrics with constant negative Ricci scalar curvature.
- 8.1. Complexity from path-integral optimization: The on-shell Liouville action is interpreted as complexity because it is proportional to the path-integral volume and reproduces the leading spatial-volume divergence.This behavior is consistent with a holographic estimate of the leading complexity divergence.
- 8.2. Holographic path-integral optimization: In holographic CFTs, maximizing the AdS Hartle–Hawking wavefunction imposes Neumann conditions on a bulk surface and yields metrics matching CFT path-integral optimization.A gravitational action with boundary and corner terms provides a finite-cutoff completion of the Liouville action.
9. Quantum complexity in quantum information theory and many-body physics
Quantum information and many-body physics use complexity to characterize operational resources, topological order, entanglement constraints, and computational structure. The review emphasizes both useful complexity measures and limits on the circuit lower bounds they can establish.
- 9. Quantum complexity in quantum information theory and many-body physics: Quantum complexity quantifies operational resources, distinguishes topological phases, and is bounded by entanglement measures under certain conditions.The section connects complexity to quantum-information tasks and many-body structure.
- 9.1. Operational tasks: Uncomplexity is the gap Cmax − C0(ρ) between maximal and actual circuit complexity, and it serves as a resource in quantum computation.For an N-qubit system, the maximal complexity scales as Cmax ∼ 2^N.
- 9.1. Operational tasks: A resource theory with noisy fuzzy gates formalizes uncomplexity extraction as preparing a state δ-close to |0⟩^⊗k using restricted operations.The setting models each implemented gate as randomly perturbed within an ϵ-ball around the intended gate.
- 9.1. Operational tasks: The complexity entropy determines the number of uncomplex qubits extractable from an arbitrary state, with matching upper and lower operational bounds.These entropy-based quantities also characterize efficiencies for tasks such as erasure and decoupling.
- 9.2–9.3. Topological order and entanglement: Topological order can impose circuit-depth lower bounds, while entanglement-based bounds require entanglement across every bipartition.Schmidt-rank methods cannot prove exponential circuit complexity because their total contribution is at best Ω(n^2).
- 9.4. Complexity from output distributions: A constant-depth preparation assumption for GHZ states conflicts with concentration constraints, and preparing |GHZ⟩ requires circuits of depth Ω(log(n)).This illustrates how output-distribution arguments can establish nontrivial circuit lower bounds.
10. Epilogue
The epilogue presents quantum complexity as a shared language linking quantum information, many-body physics, quantum gravity, and holography. It highlights quantitative progress in low-dimensional gravitational dualities while stressing unresolved higher-dimensional, de Sitter, and complexity-theoretic limitations.
- 10. Epilogue: The review surveys circuit, geometric, spreading, and tensor-network notions of complexity and relates them to thermalization, chaos, integrability, and holographic observables.It presents complexity as useful across quantum-information and gravitational settings.
- 10. Epilogue: Spread complexity in double-scaled SYK is exactly dual to the size of an Einstein–Rosen bridge in two-dimensional gravity.The review identifies this as a major quantitative achievement, while noting its narrow setting.
- 10. Epilogue: The two-dimensional result may support higher-dimensional generalization, although the simplicity of two-dimensional gravity also limits its direct scope.The review connects related two-dimensional insights to broader quantum-gravity problems.
- 10. Epilogue: A quantum dual of de Sitter spacetime remains unknown, so proposed translations between de Sitter gravity and nongravitating quantum systems are still conjectural.Possible applications to open quantum systems depend on whether such a dual exists.
- 10. Epilogue: Quantum-information tools may yield quantitative answers for how arbitrary quantum systems affect black-hole information scenarios.The review presents this as an opportunity enabled by newer resource-theoretic methods.
- 10. Epilogue: Proving superpolynomial circuit lower bounds for low-energy states of local Hamiltonians would approach separating QMA from QCMA and, consequently, P from PSPACE.This complexity-theoretic barrier makes unconditional lower bounds especially demanding.
Appendix A. Guide to acronyms
This appendix defines acronyms used throughout the paper, spanning holography, quantum information, quantum field theory, and tensor-network methods.
- AdS, BC, BCFT, CA, CAny, CFT, CV, and CV2.0 denote terms related to holography, conformal field theory, and gravitational complexity.The passage expands these abbreviations as anti-de Sitter, binding complexity, boundary conformal field theory, complexity=action, complexity=anything, conformal field theory, complexity=volume, and complexity=volume 2.0.
- cMERA, DSSYK, and MERA refer to tensor-network or many-body frameworks, while JT denotes Jackiw-Teitelboim.The listed expansions include continuous multiscale entanglement renormalization ansatz, double-scaled Sachdev-Ye-Kitaev, multiscale entanglement renormalization ansatz, and Jackiw-Teitelboim.
- ECH, EOW, ERB, EW, HRT, and OTOC denote hypotheses, boundaries, bridges, wedges, extremal-surface prescriptions, and correlation diagnostics in quantum gravity.The passage expands these as eigenstate complexity hypothesis, end-of-the-world, Einstein-Rosen bridge, entanglement wedge, Hubeny-Rangamani-Takayanagi, and out-of-time-ordered correlator.
- FS, GHZ, GUE, IR, LOCC, NLTS, QCMA, QFT, and QM identify quantum-information, condensed-matter, and field-theory concepts.The listed expansions include Fubini-Study, Greenberger-Horne-Zeilinger, Gaussian unitary ensemble, infrared, local operations and classical communication, no low-energy trivial state, quantum classical Merlin-Arthur, quantum field theory, and quantum mechanics.