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Robert C Myers - One of the best experts on this subject based on the ideXlab platform.
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Circuit Complexity for coherent states
Journal of High Energy Physics, 2018Co-Authors: Robert C Myers, Minyong Guo, Juan Hernandez, Shanming RuanAbstract:We examine the Circuit Complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]. The Complexity of the coherent states have the same UV divergences as the vacuum state Complexity and so we consider the finite increase of the Complexity of these states over the vacuum state. One observation is that generally, the optimal Circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen’s approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal Circuits, derived from these two approaches, are different.
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Circuit Complexity for free fermions
Journal of High Energy Physics, 2018Co-Authors: Lucas Hackl, Robert C MyersAbstract:We study Circuit Complexity for free fermionic field theories and Gaussian states. Our definition of Circuit Complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic Circuit Complexity, we develop a comprehensive mathematical framework to compute Circuit Complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the Circuit Complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the Complexity of excited energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.
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Circuit Complexity for free fermions
arXiv: High Energy Physics - Theory, 2018Co-Authors: Lucas Hackl, Robert C MyersAbstract:We study Circuit Complexity for free fermionic field theories and Gaussian states. Our definition of Circuit Complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic Circuit Complexity, we develop a comprehensive mathematical framework to compute Circuit Complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the Circuit Complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the Complexity of excited states. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.
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Circuit Complexity in quantum field theory
Journal of High Energy Physics, 2017Co-Authors: Robert A Jefferson, Robert C MyersAbstract:Motivated by recent studies of holographic Complexity, we examine the question of Circuit Complexity in quantum field theory. We provide a quantum Circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum Circuit model, the Complexity of the state becomes the length of the shortest geodesic in the space of Circuits. We compare the Complexity of the ground state of the free scalar field to the analogous results from holographic Complexity, and find some surprising similarities.
Toniann Pitassi - One of the best experts on this subject based on the ideXlab platform.
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Circuit Complexity proof Complexity and polynomial identity testing the ideal proof system
Journal of the ACM, 2018Co-Authors: Joshua A Grochow, Toniann PitassiAbstract:We introduce a new and natural algebraic proof system, whose Complexity measure is essentially the algebraic Circuit size of Nullstellensatz certificates. This enables us to exhibit close connections between effective Nullstellensatze, proof Complexity, and (algebraic) Circuit Complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic Circuits (VNP ≠ VP). We also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs imply the Permanent versus Determinant Conjecture. Note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any Complexity class lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems. In doing so, we highlight the importance of polynomial identity testing (PIT) in proof Complexity. In particular, we use PIT to illuminate AC0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. Finally, we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic Circuit Complexity to prove lower bounds in proof Complexity. Using the algebraic structure of our proof system, we propose a novel route to such lower bounds. Although such lower bounds remain elusive, this proposal should be contrasted with the difficulty of extending AC0[p] Circuit lower bounds to AC0[p]-Frege lower bounds.
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Circuit Complexity proof Complexity and polynomial identity testing
Foundations of Computer Science, 2014Co-Authors: Joshua A Grochow, Toniann PitassiAbstract:We introduce a new and natural algebraic proof system, which has tight connections to (algebraic) Circuit Complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic Circuits (VNP ane; VP). As a corollary, super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof Complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding Circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof Complexity.
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Circuit Complexity proof Complexity and polynomial identity testing
arXiv: Computational Complexity, 2014Co-Authors: Joshua A Grochow, Toniann PitassiAbstract:We introduce a new algebraic proof system, which has tight connections to (algebraic) Circuit Complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic Circuits (VNP is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof Complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding Circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof Complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean Circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size Circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic Circuit Complexity to prove lower bounds in proof Complexity.
Michael A Forbes - One of the best experts on this subject based on the ideXlab platform.
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proof Complexity lower bounds from algebraic Circuit Complexity
Theory of Computing, 2021Co-Authors: Michael A Forbes, Amir Shpilka, Iddo Tzameret, Avi WigdersonAbstract:We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi [26], where the Circuits comprising the proof come from various restricted algebraic Circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs whose lines are Circuits from restricted boolean Circuit classes. Essentially all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic Complexity of the polynomials (except with respect to degree, or to sparsity). Our main contributions are two general methods of converting certain algebraic lower bounds into proof Complexity ones. Both require stronger arithmetic lower bounds than common, which should hold not for a specific polynomial but for a whole family defined by it. These may be likened to some of the methods by which Boolean Circuit lower bounds are turned into related proof-Complexity ones, especially the "feasible interpolation" technique. We establish algebraic lower bounds of these forms for several explicit polynomials, against a variety of classes, and infer the relevant proof Complexity bounds. These yield separations between IPS subsystems, which we complement by simulations to create a partial structure theory for IPS systems. Our first method is a functional lower bound, a notion of Grigoriev and Razborov [25], which is a function [EQUATION] {0, 1}n → F such that any polynomial f agreeing with [EQUATION] on the boolean cube requires large algebraic Circuit Complexity. We develop functional lower bounds for a variety of Circuit classes (sparse polynomials, depth-3 powering formulas, read-once algebraic branching programs and multilinear formulas) where [EQUATION] equals [EQUATION] for a constant-degree polynomial p depending on the relevant Circuit class. We believe these lower bounds are of independent interest in algebraic Complexity, and show that they also imply lower bounds for the size of the corresponding IPS refutations for proving that the relevant polynomial p is non-zero over the boolean cube. In particular, we show super-polynomial lower bounds for refuting variants of the subset-sum axioms in these IPS subsystems. Our second method is to give lower bounds for multiples, that is, to give explicit polynomials whose all (non-zero) multiples require large algebraic Circuit Complexity. By extending known techniques, we give lower bounds for multiples for various restricted Circuit classes such sparse polynomials, sums of powers of low-degree polynomials, and roABPs. These results are of independent interest, as we argue that lower bounds for multiples is the correct notion for instantiating the algebraic hardness versus randomness paradigm of Kabanets and Impagliazzo [31]. Further, we show how such lower bounds for multiples extend to lower bounds for refutations in the corresponding IPS subsystem.
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proof Complexity lower bounds from algebraic Circuit Complexity
arXiv: Computational Complexity, 2016Co-Authors: Michael A Forbes, Amir Shpilka, Iddo Tzameret, Avi WigdersonAbstract:We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the Circuits comprising the proof come from various restricted algebraic Circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs, where proof-lines are Circuits from restricted boolean Circuit classes. Except one, all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic Complexity of the polynomials (except with respect to degree, or to sparsity). We give two general methods of converting certain algebraic lower bounds into proof Complexity ones. Our methods require stronger notions of lower bounds, which lower bound a polynomial as well as an entire family of polynomials it defines. Our techniques are reminiscent of existing methods for converting boolean Circuit lower bounds into related proof Complexity results, such as feasible interpolation. We obtain the relevant types of lower bounds for a variety of classes (sparse polynomials, depth-3 powering formulas, read-once oblivious algebraic branching programs, and multilinear formulas), and infer the relevant proof Complexity results. We complement our lower bounds by giving short refutations of the previously-studied subset-sum axiom using IPS subsystems, allowing us to conclude strict separations between some of these subsystems.
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functional lower bounds for arithmetic Circuits and connections to boolean Circuit Complexity
Conference on Computational Complexity, 2016Co-Authors: Michael A Forbes, Mrinal Kumar, Ramprasad SaptharishiAbstract:We say that a Circuit C over a field F functionally computes a polynomial P ∈ F[x1, x2, ..., xn] if for every x ∈ {0, 1}n we have that C(x) = P(x). This is in contrast to syntactically computing P, when C ≡ P as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-3 and depth-4 arithmetic Circuits for functional computation. We prove the following results: Exponential lower bounds for homogeneous depth-3 arithmetic Circuits for a polynomial in VNP. Exponential lower bounds for homogeneous depth-4 arithmetic Circuits with bounded individual degree for a polynomial in VNP. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-4 arithmetic Circuits for the Permanent imply a separation between #P and ACC0. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean Circuit Complexity. Besides, it is known from a recent result of Kumar and Saptharishi [9] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-4 Circuits imply superpolynomial lower bounds for homogeneous depth-5 Circuits. Our proofs are based on a family of new Complexity measures called shifted evaluation dimension, and might be of independent interest.
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functional lower bounds for arithmetic Circuits and connections to boolean Circuit Complexity
Electronic Colloquium on Computational Complexity, 2016Co-Authors: Michael A Forbes, Mrinal Kumar, Ramprasad SaptharishiAbstract:We say that a Circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic Circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic Circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic Circuits with bounded individual degree for a polynomial in $VNP$. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-$4$ arithmetic Circuits for the Permanent imply a separation between ${\#}P$ and $ACC$. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean Circuit Complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-$4$ Circuits imply superpolynomial lower bounds for homogeneous depth-$5$ Circuits. Our proofs are based on a family of new Complexity measures called shifted evaluation dimension, and might be of independent interest.
Joshua A Grochow - One of the best experts on this subject based on the ideXlab platform.
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Circuit Complexity proof Complexity and polynomial identity testing the ideal proof system
Journal of the ACM, 2018Co-Authors: Joshua A Grochow, Toniann PitassiAbstract:We introduce a new and natural algebraic proof system, whose Complexity measure is essentially the algebraic Circuit size of Nullstellensatz certificates. This enables us to exhibit close connections between effective Nullstellensatze, proof Complexity, and (algebraic) Circuit Complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic Circuits (VNP ≠ VP). We also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs imply the Permanent versus Determinant Conjecture. Note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any Complexity class lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems. In doing so, we highlight the importance of polynomial identity testing (PIT) in proof Complexity. In particular, we use PIT to illuminate AC0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. Finally, we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic Circuit Complexity to prove lower bounds in proof Complexity. Using the algebraic structure of our proof system, we propose a novel route to such lower bounds. Although such lower bounds remain elusive, this proposal should be contrasted with the difficulty of extending AC0[p] Circuit lower bounds to AC0[p]-Frege lower bounds.
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Circuit Complexity proof Complexity and polynomial identity testing
Foundations of Computer Science, 2014Co-Authors: Joshua A Grochow, Toniann PitassiAbstract:We introduce a new and natural algebraic proof system, which has tight connections to (algebraic) Circuit Complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic Circuits (VNP ane; VP). As a corollary, super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof Complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding Circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof Complexity.
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Circuit Complexity proof Complexity and polynomial identity testing
arXiv: Computational Complexity, 2014Co-Authors: Joshua A Grochow, Toniann PitassiAbstract:We introduce a new algebraic proof system, which has tight connections to (algebraic) Circuit Complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic Circuits (VNP is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof Complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding Circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof Complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean Circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size Circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic Circuit Complexity to prove lower bounds in proof Complexity.
Minyong Guo - One of the best experts on this subject based on the ideXlab platform.
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Circuit Complexity for generalized coherent states in thermal field dynamics
Physical Review D, 2020Co-Authors: Minyong Guo, Zhongying Fan, Jie Jiang, Xiangjing Liu, Bin ChenAbstract:In this work, we study the Circuit Complexity for generalized coherent states in thermal systems by adopting the covariance matrix approach. We focus on the coherent thermal (CT) state, which is non-Gaussian and has a nonvanishing one-point function. We find that even though the CT state cannot be fully determined by the symmetric two-point function, the Circuit Complexity can still be computed in the framework of the covariance matrix formalism by properly enlarging the covariance matrix. Now the group generated by the unitary is the semiproduct of translation and the symplectic group. If the reference state is Gaussian, the optimal geodesic is still be generated by a horizontal generator such that the Circuit Complexity can be read from the generalized covariance matrix associated to the target state by taking the cost function to be ${F}_{2}$. For a single harmonic oscillator, we discuss carefully the Complexity and its formation in the cases that the reference states are Gaussian and the target space is excited by a single mode or double modes. We show that the study can be extended to the free scalar field theory.
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Circuit Complexity for coherent states
Journal of High Energy Physics, 2018Co-Authors: Robert C Myers, Minyong Guo, Juan Hernandez, Shanming RuanAbstract:We examine the Circuit Complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]. The Complexity of the coherent states have the same UV divergences as the vacuum state Complexity and so we consider the finite increase of the Complexity of these states over the vacuum state. One observation is that generally, the optimal Circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen’s approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal Circuits, derived from these two approaches, are different.