The Equivalence of Sampling and Searching

Aaronson, Scott

doi:10.1007/978-3-642-20712-9_1

Cited by 11 publications

(21 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our results show that any family of quantum circuits that both admits a classical poly-box and satisfies the poly-sparsity condition is in the complexity class SampP [34]. In light of the multiple known constructions of poly-boxes (see Refs.…”

Section: B Discussionmentioning

confidence: 88%

From estimation of quantum probabilities to simulation of quantum circuits

Pashayan,

Bartlett,

Gross

2017

Preprint

View full text Add to dashboard Cite

We show that there are two distinct obstacles to an efficient classical simulation of a general quantum circuit. The first obstacle, which has been well-studied, is the difficulty of efficiently estimating the probability associated with a particular measurement outcome. We show that this alone does not determine whether a quantum circuit can be efficiently simulated. The second obstacle is that, in general, there can be an exponential number of 'relevant' outcomes that are needed for an accurate simulation, and so efficient simulation may not be possible even in situations where the probabilities of individual outcomes can be efficiently estimated. We show that these two obstacles are distinct, overcoming the former obstacle being necessary but not sufficient for simulability whilst overcoming the pair is jointly sufficient for simulability. Specifically, we prove that a family of quantum circuits is efficiently simulable if it satisfies two properties: one related to the efficiency of Born rule probability estimation, and the other related to the sparsity of the outcome distribution. We then prove a pair of hardness results (using standard complexity assumptions and a variant of a commonly-used average case hardness conjecture), where we identify families of quantum circuits that satisfy one property but not the other, and for which efficient simulation is not possible. To prove our results, we consider a notion of simulation of quantum circuits that we call -simulation. This notion is less stringent than exact sampling and is now in common use in quantum hardness proofs.

show abstract

Section: B Discussionmentioning

confidence: 88%

From estimation of quantum probabilities to simulation of quantum circuits

Pashayan,

Bartlett,

Gross

2017

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently, and directly motivated by the present work, Aaronson [4] proved a general connection between sampling problems and search problems.…”

Section: Sampling and Search Problemsmentioning

confidence: 99%

“…However, while Aaronson [4] proved an extremely general connection between search problems and sampling problems, that connection only works for approximate sampling, not exact sampling.…”

Section: The Exact Casementioning

confidence: 99%

See 1 more Smart Citation

Untitled

Aaronson¹,

2013

Self Cite

View full text Add to dashboard Cite

“…That is, a valid classical simulation of the quantum circuit can return any measurement outcome a genuine quantum device may output with nonzero probability. 1 Unfortunately, we believe 2 that proving a separation against NC 1 under this model will require new nontrivial techniques in circuit complexity. Instead, we introduce interactivity into our model as a way to circumvent these challenges.…”

Section: Introductionmentioning

confidence: 99%

Interactive shallow Clifford circuits: quantum advantage against NC$^1$ and beyond

Grier¹,

Schaeffer²

2019

Preprint

View full text Add to dashboard Cite

Recent work of Bravyi et al. and follow-up work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constant-depth quantum circuits can perform a task which constant-depth classical (i.e., AC 0 ) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constant-depth classical circuits are too weak to yield a convincing real-world demonstration of quantum advantage. We attempt to hold on to the advantages of the above results, while increasing the power of the classical model.Our main result is a two-round interactive task which is solved by a constant-depth quantum circuit (using only Clifford gates, between neighboring qubits of a 2D grid, with Pauli measurements), but such that any classical solution would necessarily solve ⊕L-hard problems. This implies a more powerful class of constant-depth classical circuits (e.g., AC 0 [p] for any prime p) unconditionally cannot perform the task. Furthermore, under standard complexity-theoretic conjectures, log-depth circuits and log-space Turing machines cannot perform the task either.Using the same techniques, we prove hardness results for weaker complexity classes under more restrictive circuit topologies. Specifically, we give QNC 0 interactive tasks on 2 × n and 1 × n grids which require classical simulations of power NC 1 and AC 0 [6], respectively. Moreover, these hardness results are robust to a small constant fraction of error in the classical simulation.We use ideas and techniques from the theory of branching programs, quantum contextuality, measurement-based quantum computation, and Kilian randomization.

show abstract

The Equivalence of Sampling and Searching

Cited by 11 publications

References 9 publications

From estimation of quantum probabilities to simulation of quantum circuits

From estimation of quantum probabilities to simulation of quantum circuits

Untitled

Interactive shallow Clifford circuits: quantum advantage against NC$^1$ and beyond

Contact Info

Product

Resources

About