Non-invertible symmetries define quantum gates with generalized complexity distances, and simple objects in symmetry categories turn out to be computationally complex in concrete 4D and 2D QFT examples.
The geometry of quantum computation
11 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.
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Open-system quantum complexity is governed by a sub-Finslerian geometry whose curvature depends on the cost penalties for unitary and dissipative controls.
Orbifolds of N=4 SYM produce SCFTs whose dilatation operator in a subsector is realized by a tunable spin chain whose eigenvalue statistics exhibit chaos for specific marginal couplings.
A geometric formalism yields tight quantum speed limits for quantum gates by mapping unitary evolution to minimal-length curves with curvature bounds.
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.
Presents a universal parametrized quantum circuit ansatz based on Euler-Cartan decompositions, benchmarked on energy spectra of lattice QFT models with short- and long-range interactions.
Studies holographic complexity in the Klebanov-Strassler background, reporting common scaling with confinement scale across functionals and more complex UV divergences than in AdS.
New combinatorial proofs and circuit designs for quantum error correction reduce physical qubit overhead by up to 10x and time overhead by 2-6x for codes including Steane, Golay, and surface codes.
Geometric complexity of physical maps is bounded below by execution error, forcing divergent resources for zero-error state resets in both classical and quantum settings.
A timelike quantum focusing conjecture implies a complexity-based quantum strong energy condition and a complexity bound analogous to the covariant entropy bound for suitable codimension-0 field theory complexity measures.
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.
citing papers explorer
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Generalized Complexity Distances and Non-Invertible Symmetries
Non-invertible symmetries define quantum gates with generalized complexity distances, and simple objects in symmetry categories turn out to be computationally complex in concrete 4D and 2D QFT examples.
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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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Controlled Chaos in 4D SCFTs
Orbifolds of N=4 SYM produce SCFTs whose dilatation operator in a subsector is realized by a tunable spin chain whose eigenvalue statistics exhibit chaos for specific marginal couplings.
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How fast can a quantum gate be? Exact speed limits from geometry
A geometric formalism yields tight quantum speed limits for quantum gates by mapping unitary evolution to minimal-length curves with curvature bounds.
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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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Universal Euler-Cartan Circuits for Quantum Field Theories
Presents a universal parametrized quantum circuit ansatz based on Euler-Cartan decompositions, benchmarked on energy spectra of lattice QFT models with short- and long-range interactions.
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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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Lower overhead fault-tolerant building blocks for noisy quantum computers
New combinatorial proofs and circuit designs for quantum error correction reduce physical qubit overhead by up to 10x and time overhead by 2-6x for codes including Steane, Golay, and surface codes.
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Geometric complexity in thermodynamics
Geometric complexity of physical maps is bounded below by execution error, forcing divergent resources for zero-error state resets in both classical and quantum settings.
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A Timelike Quantum Focusing Conjecture
A timelike quantum focusing conjecture implies a complexity-based quantum strong energy condition and a complexity bound analogous to the covariant entropy bound for suitable codimension-0 field theory complexity measures.
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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.