Extractors for Circuit Sources

Viola, Emanuele

doi:10.1137/11085983x

Cited by 31 publications

(21 citation statements)

References 46 publications

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“…A powerful tool that might be used to address this question is given in Ref. [37]. In particular, the author shows that any distribution over n-bit strings with linear min-entropy (i.e., Θ(n)) which is generated by applying a constant-depth classical circuit to a uniformly random input string can be expressed as a convex combination of simple distributions ("bit-block sources") with linear min-entropy.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Quantum advantage with shallow circuits

Bravyi¹,

Gosset²,

König

2018

Science

347

313

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We prove that constant-depth quantum circuits are more powerful than their classical counterparts. To this end we introduce a non-oracular version of the Bernstein-Vazirani problem which we call the 2D Hidden Linear Function problem. An instance of the problem is specified by a quadratic form q that maps n-bit strings to integers modulo four. The goal is to identify a linear boolean function which describes the action of q on a certain subset of n-bit strings. We prove that any classical probabilistic circuit composed of bounded fan-in gates that solves the 2D Hidden Linear Function problem with high probability must have depth logarithmic in n. In contrast, we show that this problem can be solved with certainty by a constant-depth quantum circuit composed of one-and two-qubit gates acting locally on a two-dimensional grid.building on earlier studies of so-called IQP circuits [6,7,8], gave further evidence that the output distribution of SQCs may be hard to sample classically even if a constant statistical error is allowed; see also Ref. [9]. A more powerful model of computation consisting of logarithmic-depth quantum circuits assisted by polynomial-time classical computation is known to be capable of solving hard problems such as factoring [10].Parallelism and circuit depth are important considerations when designing quantum algorithms that can be executed in the near-future on small quantum computers that may lack error correction capabilities [11,12,13]. While the overhead for encoding and manipulating quantum data fault-tolerantly is asymptotically small [14,15], it remains prohibitive for current technology. A quantum computation without error correction can only compute for a constant amount of time before the qubits decohere and the entropy builds up [16]. In this situation one may wish to parallelize the computation as much as possible to fit within the coherence time.What can we hope to prove concerning the computational power of SQCs ? A rigorous proof that SQCs outperform polynomial-time classical algorithms for some computational task is arguably beyond our reach (as it would imply a separation between the complexity classes BQP and BPP). In this paper we set a less ambitious goal and pose the following question:Can constant-depth quantum circuits solve a computational problem that constant-depth classical circuits cannot?

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Section: Conclusion and Open Problemsmentioning

confidence: 99%

Quantum advantage with shallow circuits

Bravyi¹,

Gosset²,

König

2018

Science

347

313

View full text Add to dashboard Cite

show abstract

“…Average-case reductions from NC 0 to AC 0 have previously appeared in the literature (cf. [Vio14]), based on the original switching lemma [Hås86]. In this paper, we will use multi-switching lemmas, which handle multiple output circuits much better, and were recently proved by Håstad [Hås14] and Rossman [Ros17].…”

Section: From Nc 0 Circuits To Ac 0 Circuitsmentioning

confidence: 99%

Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

Watts

Kothari

Schaeffer

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Recently, Bravyi, Gosset, and König (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC 0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC 0 circuits). In other words, they exhibited a search problem in QNC 0 that is not in NC 0 .We strengthen their result by proving that the 2D HLF problem is not contained in AC 0 , the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC 0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC 0 , which we call the Relaxed Parity Halving Problem, which is easier to work with. We prove our AC 0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem.As a step towards even stronger lower bounds, we present a search problem that we call the Parity Bending Problem, which is in QNC 0 /qpoly (QNC 0 circuits that are allowed to start with a quantum state of their choice that is independent of the input), but is not even in AC 0 [2] (the class AC 0 with unbounded fan-in XOR gates).All the quantum circuits in our paper are simple, and the main difficulty lies in proving the classical lower bounds. For this we employ a host of techniques, including a refinement of Håstad's switching lemmas for multi-output circuits that may be of independent interest, the Razborov-Smolensky AC 0 [2] lower bound, Vazirani's XOR lemma, and lower bounds for non-local games. * Massachusetts Institute of Technology. abenewat@mit.edu.

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“…But our techniques seem unable to establish this, and more generally we raise the question of proving such a statistical bound for any explicit distribution. We mention that after this work, and using the ideas in it, there has been progress on proving sampling lower bounds on AC 0 circuits, see Viola (2011).…”

Section: A Subset C ⊂ {0 1}mentioning

confidence: 99%

Bounded-Depth Circuits Cannot Sample Good Codes

Lovett

Viola

2012

comput. complex.

Self Cite

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Abstract. We study a variant of the classical circuit-lower-bound problems: proving lower bounds for sampling distributions given random bits. We prove a lower bound of 1 − 1/n Ω(1) on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC 0 ) circuit f : {0, 1} poly(n) → {0, 1} n , and (ii) the uniform distribution over any code C ⊆ {0, 1} n that is "good", i.e. has relative distance and rate both Ω(1). This seems to be the first lower bound of this kind. We give two simple applications of this result: (1) any data structure for storing codewords of a good code C ⊆ {0, 1} n requires redundancy Ω(log n), if each bit of the codeword can be retrieved by a small AC 0 circuit; (2) for some choice of the underlying combinatorial designs, the output distribution of Nisan's pseudorandom generator against AC 0 circuits of depth d cannot be sampled by small AC 0 circuits of depth less than d.

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Extractors for Circuit Sources

Cited by 31 publications

References 46 publications

Quantum advantage with shallow circuits

Quantum advantage with shallow circuits

Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

Bounded-Depth Circuits Cannot Sample Good Codes

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