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Quantum Computation as Geometry
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Quantum computers hold great promise, but it remains a challenge to find efficient quantum circuits that solve interesting computational problems. We show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms, or to prove limitations on the power of quantum computers.
Forward citations
Cited by 16 Pith papers
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The Geometry of Quantum Complexity in Open Systems
Open-system quantum complexity is governed by a sub-Finslerian geometry whose curvature depends on the cost penalties for unitary and dissipative controls.
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Complexity Inequalities for Quantum Subsystems
Introduces tripartite complexity and complexity gap for three-region subsystems and reports that the gap has a definite sign in holographic volume complexity, Fisher-Rao Gaussian complexity, and Krylov-space approaches.
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Complexity measures in QFT and constrained geometric actions
The authors claim to rule out inhomogeneous complexity costs such as F_kappa and F_sigma^2 and to single out F_⟨H^2⟩ as the canonical complexity measure, but the no-go proof is incomplete.
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Complexity measures in holographic cascading theories with multiscale dynamics
In holographic B8 gauge theories, complexity growth flattens near a walking conformal regime, while Krylov oscillation periods track the infrared scale and persist in screened, non-confining phases.
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Explicit Matrices over $\mathbb Z_2$ with CNOT and Row Complexity $4n-\mathrm{o}(n)$ and Local Logic Gates
Explicit n imes n matrices over Z_2 require 4n−o(n) CNOT/row/2-local linear gates, and the same bound holds for the quantum complexity of the associated affine permutations.
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Page transition for the complexity of an evaporating black hole
The complexity of radiation from an evaporating black hole is argued to undergo a sharp Page-like transition, dominated after the Page time by the volume of an island in the entanglement wedge.
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Quantum Cosmology in Krylov Space: Complexity and Entropy
In a sharply peaked Gaussian state of a flat FLRW universe with a massless scalar clock, Krylov state complexity grows as σ²(φ−φ0)²/4 and operator complexity is exactly twice that, in both Wheeler-DeWitt and loop quan...
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Spread complexity and the saturation of wormhole size
For finite-N DSSYK, the chord basis is the early part of the physical Krylov basis, and spread complexity, taken as the non-perturbative ER bridge size, saturates after a universality-class-dependent peak and slope.
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Towards the Web of Quantum Chaos Diagnostics
Haar-averaged higher-point out-of-time-order correlators equal multi-fold Loschmidt echoes, and complexity is conjectured to satisfy C^2 ~ -log LE.
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Emergence of Krylov complexity through quantum walks: An exploration of the quantum origins of complexity
Reducing a graph walk to distance-layers reproduces Krylov/spread complexity, yielding analytic finite-q SYK Lanczos coefficients and hypercube complexity D sin²(t/D), with faster saturation than classical-walk circuits.
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Generalized CV Conjecture and Krylov Complexity in Two-Mode Hermitian Systems via Information Geometry
Krylov complexity equals Fubini-Study volume for closed and open two-mode squeezed states, providing analytic support for the generalized CV conjecture via information geometry.
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Holographic complexity of the Klebanov-Strassler background
Studies holographic complexity in the Klebanov-Strassler background, reporting common scaling with confinement scale across functionals and more complex UV divergences than in AdS.
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Post-Selection Probability and Fidelity of Bidirectional Teleportation
Post-selection probability and fidelity of bidirectional teleportation are expressed via the Loschmidt echo, revealing initial-state dependence of fidelity and stability of probability in integrable models.
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Nielsen complexity with multiple cost factors
Generalizes Nielsen complexity to multiple cost factors, derives modified Euler-Arnold and Jacobi equations, and examines effects on conjugate points in single-qubit and SYK systems.
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Wavefunction branches demand a definition!
A perspective comparing two quantum-complexity definitions of wavefunction branches, finding neither satisfactory and identifying the open problems that remain.
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Holographic entanglement entropy and complexity for the cosmological braneworld model
Time-dependent holographic entanglement entropy and complexity are computed perturbatively for braneworld FLRW universes with radiation, matter, and exotic matter by using time-dependent brane positions in black brane...
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