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arXiv:2311.04277 · Quantum

The Complexity of Being Entangled

Stefano Baiguera, Shira Chapman, Giuseppe Policastro, Tal Schwartzman

Published Sep 12, 2024• 5 citations
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Abstract

Nielsen's approach to quantum state complexity relates the minimal number of quantum gates required to prepare a state to the length of geodesics computed with a certain norm on the manifold of unitary transformations. For a bipartite system, we investigate binding complexity, which corresponds to norms in which gates acting on a single subsystem are free of cost. We reduce the problem to the study of geodesics on the manifold of Schmidt coefficients, equipped with an appropriate metric. Binding complexity is closely related to other quantities such as distributed computing and quantum communication complexity, and has a proposed holographic dual in the context of AdS/CFT. For finite dimensional systems with a Riemannian norm, we find an exact relation between binding complexity and the minimal Rényi entropy. We also find analytic results for the most commonly used non-Riemannian norm (the so-called F1 norm) and provide lower bounds for the associated notion of state complexity ubiquitous in quantum computation and holography. We argue that our results are valid for a large class of penalty factors assigned to generators acting across the subsystems. We demonstrate that our results can be borrowed to study the usual complexity (not-binding) for a single spin for the case of the F1 norm which was previously lacking from the literature. Finally, we derive bounds for multi-partite binding complexities and the related (continuous) circuit complexity where the circuit contains at most 2 -local interactions.

DOI
10.22331/q-2024-09-12-1472

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Nielsen's approach to quantum state complexity relates the minimal number of quantum gates required to prepare a state to the length of geodesics computed with a certain norm on the manifold of unitary transformations. For a bipartite system, we investigate binding complexity, which corresponds to norms in which gates acting on a single subsystem are free of cost. We reduce the problem to the study of geodesics on the manifold of Schmidt coefficients, equipped with an appropriate metric. Binding complexity is closely related to other quantities such as distributed computing and quantum communication complexity, and has a proposed holographic dual in the context of AdS/CFT. For finite dimensional systems with a Riemannian norm, we find an exact relation between binding complexity and the minimal Rényi entropy. We also find analytic results for the most commonly used non-Riemannian norm (the so-called F1 norm) and provide lower bounds for the associated notion of state complexity ubiquitous in quantum computation and holography. We argue that our results are valid for a large class of penalty factors assigned to generators acting across the subsystems. We demonstrate that our results can be borrowed to study the usual complexity (not-binding) for a single spin for the case of the F1 norm which was previously lacking from the literature. Finally, we derive bounds for multi-partite binding complexities and the related (continuous) circuit complexity where the circuit contains at most 2 -local interactions.
We introduce “binding complexity”, a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show that our states can be prepared by the Euclidean path integral in (0 + 1)-dimensional quantum mechanics on graphs with wormhole-like structure. We compute the binding complexity of our states by adapting the Euler-Arnold approach to Nielsen’s geometrization of gate counting, and find a scaling with entropy that resembles a result for the interior volume of holographic multiboundary wormholes. We also compute the binding complexity of general coherent states in perturbation theory, and show that for “double-trace deformations” of the Hamiltonian the effects resemble expansion of a wormhole interior in holographic theories.
Abstract We introduce “binding complexity”, a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show that our states can be prepared by the Euclidean path integral in (0 + 1)-dimensional quantum mechanics on graphs with wormhole-like structure. We compute the binding complexity of our states by adapting the Euler-Arnold approach to Nielsen’s geometrization of gate counting, and find a scaling with entropy that resembles a result for the interior volume of holographic multiboundary wormholes. We also compute the binding complexity of general coherent states in perturbation theory, and show that for “double-trace deformations” of the Hamiltonian the effects resemble expansion of a wormhole interior in holographic theories.
Computational complexity is a quantum information concept that recently has found applications in the holographic understanding of the black hole interior. We consider quantum computational complexity for nnn qubits using Nielsen's geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parametrizes the cost of the elementary computational gates. In order to reproduce desired features in holography, such as ergodicity and exponential maximal complexity for large number of qubits nnn, negative curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular sectional curvatures in the large nnn limit. We investigate a choice of penalties in which we can obtain negative curvatures in a smooth way. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.
A bstract Quantum complexity of conformal field theory (CFT) states has recently gained significant attention, both as a diagnostic tool in condensed matter systems and in connection with holographic observables probing black hole interiors. Previous studies have primarily focused on cases where all generators of the conformal group contribute equally to the cost of building a circuit. In this work, we present a general framework for studying the complexity of circuits in generic Lie groups, where penalty factors assign relative weights to different generators. Our approach constructs a metric on the coset space of quantum states, induced from a (pseudo-)Riemannian norm on the space of unitary circuits. The geodesics of this metric are interpreted as optimal circuits. The method builds on the formalism of (pseudo-)Riemannian submersions and connects naturally to other prescriptions in the literature, including cost function minimization along stabilizer directions and constructions based on coadjoint orbits. As a concrete application, we compute state complexity for states in one- and two-dimensional CFTs. For specific choices of penalty factors, our prescription yields a positive-definite metric with a viable interpretation as complexity; in other cases, the resulting metric is indefinite. In the viable regime, we derive analytic results when a specific penalty factor is turned off, develop perturbative expansions for small values of the penalty factors, and provide numerical results in the general case. We comment on the relation of our measure of complexity to holography.
Abstract Motivated by recent studies of quantum computational complexity in quantum field theory and holography, we discuss how weighting certain classes of gates building up a quantum circuit more heavily than others affects the complexity. Utilizing Nielsen’s geometric approach to circuit complexity, we investigate the effects for a regulated field theory for which the optimal circuit is a representation of GL(N,R)GL(N,\mathbb{R})GL(N,R). More precisely, we work out how a uniformly chosen weighting factor acting on the entangling gates affects the complexity and, particularly, its divergent behavior. We show that assigning a higher cost to the entangling gates increases the complexity. Employing penalized and unpenalized complexities for the Fκ=2\mathcal{F}_{\kappa=2}Fκ=2​ cost, we further find an interesting relation between the latter and that based on the unpenalized Fκ=1\mathcal{F}_{\kappa=1}Fκ=1​ cost. In addition, we exhibit how imposing such penalties modifies the leading-order UV divergence in the complexity. We show that appropriately tuning the gate weighting eliminates the additional logarithmic factor, thus resulting in a simple power-law scaling. We also compare the circuit complexity with holographic predictions, specifically based on the complexity=action conjecture, and relate the weighting factor to certain bulk quantities. Finally, we comment on certain expectations concerning the role of gate penalties in defining complexity in field theory and also speculate on possible implications for holography.
We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form su(1,1) algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals.