The PCP theorem by gap amplification

Dinur, Irit

doi:10.1145/1132516.1132553

Cited by 217 publications

(384 citation statements)

References 23 publications

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“…(7) or Eq. (6), that the trace tr(U (s 1 )U (s 2 )...U (s m )) can be reduced to a trivial trace of the identity matrix by canceling successive appearances of U (s)U (s + D/2) and replacing them with 1 1. The contribution of such choices to E 1 is proportional to a return probability of a random walk on a Cayley tree as will be seen.…”

Section: Lower Bounds and Numerical Resultsmentioning

confidence: 99%

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Random unitaries give quantum expanders

Hastings

2007

Phys. Rev. A

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We show that randomly choosing the matrices in a completely positive map from the unitary group gives a quantum expander. We consider Hermitian and non-Hermitian cases, and we provide asymptotically tight bounds in the Hermitian case on the typical value of the second largest eigenvalue. The key idea is the use of Schwinger-Dyson equations from lattice gauge theory to efficiently compute averages over the unitary group.Recently, two papers [1, 2] introduced the idea of expander maps: quantum analogues of expander graphs. An expander graph[4] may be defined in several ways. One is the property of having a large number of vertices, a small coordination number for each vertex, and also having a gap in the spectrum of the diffusion equation on the graph, so that a particle classically diffusing on an expander graph rapidly loses memory of where it started. In the quantum case, we replace the random process of diffusion by a completely positive, trace preserving map E(M ). We define a quantum expander to be such a map from the space of N -by-N matrices M to the same space with the following properties. First, N is large, in analogy to the large number of vertices. Second, the map has eigenvalues λ 1 , λ 2 , λ 3 , ..., λ N 2 , with λ 1 = 1 and |λ a | ≤ 1 − δ for all a > 1 so that the eigenvalue spectrum has a gap. Finally, the map can be written asfor some relatively small value of D, with These maps were applied in [1] to construct many-body states in one dimension with the property of having a short correlation length (this corresponds to the gap δ in the spectrum of eigenvalues of E), small Hilbert space dimension on each site (this corresponds to the small D), and yet large entanglement entropy (this corresponds to the large entropy of the eigenvector of E with unit eigenvalue). Since expander graphs have a large number of applications in problems dealing with classical statistics, such as in error-correcting codes [5], derandomization, and the PCP theorem [6], to name a few, it seems worth further exploring the quantum case.A number of possible forms of an expander map are possible: in [2] an expander was defined as havingfor some unitary matrices U (s), and in fact this is the form of A(s) considered in this paper. However, the more general definition with arbitrary A(s) constrained byA(s)A † (s) = 1 1 seems also useful; in fact, although we do not consider it in this paper, it may be useful to weaken this constraint further, and explore the properties of completely positive, trace preserving maps, with no other constraint on the A, requiring only that the entropy of the density matrix which is the eigenvector with unit eigenvalue is large [3].Our goal is to try to find families of maps with arbitrarily large N , such that the gap δ is bounded below by some N -independent constant and such that D does not grow too rapidly with N . The first paper[1] provided an explicit construction of such a family of maps with D of order log(N ) and provided numerical evidence for an alternate construction with D independent ...

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Section: Lower Bounds and Numerical Resultsmentioning

confidence: 99%

“…Since expander graphs have a large number of applications in problems dealing with classical statistics, such as in error-correcting codes [5], derandomization, and the PCP theorem [6], to name a few, it seems worth further exploring the quantum case.…”

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confidence: 99%

Random unitaries give quantum expanders

Hastings

2007

Phys. Rev. A

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“…Thus, even in light of recent advances in the computational efficiency of PCPs [BS08,Din07,MR08,BCGT13b], it seems wise to first investigate efficient implementations of SNARGs in the preprocessing model, which is a less demanding model because it allows G to conduct a one-time expensive computation "as a setup phase". Despite the expensive preprocessing, this model is potentially useful for many applications: while the generator G does require a lot of work to set up the system's public parameters (which only depend on the given circuit C but not the input to C), this work can be subsequently amortized over many succinct proof verifications (where each proof is with respect to a new, adaptively-chosen, input to C).…”

Section: Motivationmentioning

confidence: 99%

SNARKs for C: Verifying Program Executions Succinctly and in Zero Knowledge

Ben–Sasson

Chiesa²,

Genkin

et al. 2013

Lecture Notes in Computer Science

405

259

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Abstract. An argument system for NP is a proof system that allows efficient verification of NP statements, given proofs produced by an untrusted yet computationally-bounded prover. Such a system is non-interactive and publiclyverifiable if, after a trusted party publishes a proving key and a verification key, anyone can use the proving key to generate non-interactive proofs for adaptivelychosen NP statements, and proofs can be verified by anyone by using the verification key.We present an implementation of a publicly-verifiable non-interactive argument system for NP. The system, moreover, is a zero-knowledge proof-ofknowledge. It directly proves correct executions of programs on TinyRAM, a nondeterministic random-access machine tailored for efficient verification. Given a program P and time bound T , the system allows for proving correct execution of P , on any input x, for up to T steps, after a one-time setup requiringÕ(|P |·T ) cryptographic operations. An honest prover requiresÕ(|P |·T ) cryptographic operations to generate such a proof, while proof verification can be performed with only O(|x|) cryptographic operations. This system can be used to prove the correct execution of C programs, using our TinyRAM port of the GCC compiler.This yields a zero-knowledge Succinct Non-interactive ARgument of Knowledge (zk-SNARK) for program executions, in the preprocessing model -a powerful solution for delegating NP computations, with several features not achieved by previously-implemented primitives.Our approach builds on recent theoretical progress in the area. We present efficiency improvements and implementations of two main ingredients: 1. Given a C program, we produce a circuit whose satisfiability encodes the correctness of execution of the program. Leveraging nondeterminism, the generated circuit's size is merely quasilinear in the size of the computation. In particular, we efficiently handle arbitrary and data-dependent loops, control flow, and memory accesses. This is in contrast with existing "circuit generators", which in the general case produce circuits of quadratic size. 2. Given a linear PCP for verifying satisfiability of circuits, we produce a corresponding SNARK. We construct such a linear PCP (which, moreover, is zero-knowledge and very efficient) by building and improving on recent work on quadratic arithmetic programs.

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“…Locally testable codes have implicitly been a subject of active study ever since the work of Blum, Luby, and Rubinfeld [1990] that showed that (effectively) the Hadamard code is 3-locally testable. They play a major role in the construction of PCPs Safra, 1998, Arora et al, 1998] from the early days of this theorem and continuing through the recent work of Dinur [2007]. Their systematic investigation was started in [Goldreich and Sudan, 2006] and yet most basic questions about their limits remain unanswered (e.g., is there an asymptotically good family of locally testable codes?…”

Section: Locally Correctable and Locally Testable Codes Affine-invarmentioning

confidence: 99%

Limits on the Rate of Locally Testable Affine-Invariant Codes

Ben–Sasson¹,

Sudan²

2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, "algebraic property testing" is not wellunderstood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "affine-invariant properties", i.e., properties that are closed under summation, and under affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its affine-invariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of affine-invariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of Reed-Muller codes (the most popular form of locally testable codes, which also happen to be affine-invariant) could be found by systematically exploring the space of affine-invariant properties.In this work we rule out this possibility and show that general (linear) affine-invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity. A central impediment to proving such results was the limited understanding of the structural restrictions on affine-invariant properties imposed by the existence of local tests. We manage to overcome this limitation and present a clean restriction satisfied by affine-invariant properties that exhibit local tests. We do so by relating the problem to that of studying the set of solutions of a certain nice class of (uniform, homogenous, diagonal) systems of multivariate polynomial equations. Our main technical result completely characterizes (combinatorially) the set of zeroes, or algebraic set, of such systems.

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The PCP theorem by gap amplification

Cited by 217 publications

References 23 publications

Random unitaries give quantum expanders

Random unitaries give quantum expanders

SNARKs for C: Verifying Program Executions Succinctly and in Zero Knowledge

Limits on the Rate of Locally Testable Affine-Invariant Codes

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