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Algebrization: A new barrier in complexity theory
 MIT Theory of Computing Colloquium
, 2007
"... Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a ..."
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Cited by 51 (3 self)
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Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a lowdegree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1
Circuit lower bounds for MerlinArthur classes
 In Proc. ACM STOC
, 2007
"... We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ..."
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Cited by 16 (1 self)
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We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the averagecase setting, i.e., we show that MA/1 cannot be solved on more than 1/2 + 1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudorandom generators computable using O(1) bits of advice. 1
A Full Characterization of Quantum Advice
"... We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of comp ..."
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We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly ⊆ QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly ⊆ PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools—including a result of Alon et al. on learning of realvalued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on ‘QMA+ superverifiers’—and also creating some new ones. The main new tool is a socalled majoritycertificates lemma, which is related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f ∈ S can be expressed as the pointwise majority of m = O (n) functions f1,..., fm ∈ S, such that each fi is the unique function in S compatible with O (log S) input/output constraints.
MIT
"... Abstract Any proof of P 6 = NP will have to overcome two barriers: relativization and natural proofs.Yet over the last decade, we have seen circuit lower bounds (for example, that PP does nothave linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether ther ..."
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Abstract Any proof of P 6 = NP will have to overcome two barriers: relativization and natural proofs.Yet over the last decade, we have seen circuit lower bounds (for example, that PP does nothave linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory.In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give thesimulating machine access not only to an oracle A, but also to a lowdegree extension of A overa finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetizationboth inclusions such as IP = PSPACE and MIP = NEXP, andseparations such as MAEXP 6ae P/polydo indeed algebrize. Second, we show that almost all ofthe major open problemsincluding P versus NP, P versus RP, and NEXP versus P/polywillrequire nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP.Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some ofour lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O (pn log n)communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL 6 = NP.
Research Statement
, 2007
"... Most of my research deals with two questions: first, what are the ultimate limits on what can feasibly be computed in the physical world? Second, how can studying those limits shed light on basic issues in physics and cosmology? The first question involves bringing physics into computational complex ..."
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Most of my research deals with two questions: first, what are the ultimate limits on what can feasibly be computed in the physical world? Second, how can studying those limits shed light on basic issues in physics and cosmology? The first question involves bringing physics into computational complexity theory; the second, bringing computational complexity theory into physics. A priori, one might wonder whether there is any useful bridge to be built between these two subjects, one sturdy enough to carry not just metaphors but nontrivial technical results. As it happens, such a bridge has existed for thirteen years. Ever since I learned how to program, I had imagined that the physical world consists of “bits all the way down. ” It seemed obvious to me that, if only we could probe nature at the Planck scale, we would find nothing but a vast array of bits getting updated by simple local rules: Conway’s Game of Life writ large. The specific form of the rules was of no great consequence, since according to the Extended ChurchTuring Thesis, any “reasonable ” set of rules can simulate any other with at most polynomial slowdown. After Peter Shor discovered his factoring algorithm [50], I and others who thought similarly were faced with a choice. Either (1) the Extended ChurchTuring Thesis is false, (2) quantum mechanics as conventionally understood is false, or
MIT
"... We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of comp ..."
Abstract
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We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly ⊆ QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly ⊆ PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools—including a result of Alon et al. on learning of realvalued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on ‘QMA+ superverifiers’—and also creating some new ones. The main new tool is a socalled majoritycertificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f ∈ S can be expressed as the pointwise