Simulating quantum computers with probabilistic methods

Nest, Maarten Van den

doi:10.48550/arxiv.0911.1624

Cited by 4 publications

(7 citation statements)

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“…where in the last line we used Eq. (20). The second term above can be computed using a repeated application of Eq.…”

Section: Theorem 4 ([6]mentioning

confidence: 99%

“…n such that Φ = β|α . We claim that the states |α and |β are computationally tractable [20] in the sense that their amplitudes in the standard basis are easy to compute classically. Indeed, consider some amplitude x|α .…”

Section: Introductionmentioning

confidence: 98%

“…A slight generalization of the above argument shows that a classical computer can efficiently sample the probability distribution P (x) ≡ | x|α | 2 . Now the forrelation Φ = β|α can be approximated by a Monte Carlo method due to Van den Nest [20] which is based on the identity…”

Section: Introductionmentioning

confidence: 99%

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Classical algorithms for Forrelation

Bravyi¹,

Gosset²,

Grier³

et al. 2021

Preprint

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We study the forrelation problem: given a pair of n-bit boolean functions f and g, estimate the correlation between f and the Fourier transform of g. This problem is known to provide the largest possible quantum speedup in terms of its query complexity and achieves the landmark oracle separation between the complexity class BQP and the Polynomial Hierarchy. Our first result is a classical algorithm for the forrelation problem which has runtime O(n2 n/2 ). This is a nearly quadratic improvement over the best previously known algorithm. Secondly, we introduce a graph-based forrelation problem where n binary variables live at vertices of some fixed graph and the functions f, g are products of terms dscribing interactions between nearest-neighbor variables. We show that the graph-based forrelation problem can be solved on a classical computer in time O(n 2 ) for any bipartite graph, any planar graph, or, more generally, any graph which can be partitioned into two subgraphs of constant treewidth. The graphbased forrelation is simply related to the variational energy achieved by the Quantum Approximate Optimization Algorithm (QAOA) with two entangling layers and Isingtype cost functions. By exploiting the connection between QAOA and the graph-based forrelation we were able to simulate the recently proposed Recurisve QAOA with two entangling layers and 225 qubits on a laptop computer.

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“…where in the last line we used Eq. (20). The second term above can be computed using a repeated application of Eq.…”

Section: Theorem 4 ([6]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Classical algorithms for Forrelation

Bravyi¹,

Gosset²,

Grier³

et al. 2021

Preprint

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show abstract

“…If there are only polynomial number of samples output by the quantum device and the output probability distribution only has non-negligible support on polynomial number of strings, then by the Chernoff?Hoeffding bound, one can only obtain information about the probabilities with polynomial precision [26]. Therefore it is enough to have a weak simulator that approximates the probabilities to the following degree:…”

Section: Appendix A: Notions Of Weak Simulationmentioning

confidence: 99%

Feynman-path type simulation using stabilizer projector decomposition of unitaries

Huang,

Love

2020

Preprint

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We propose a classical simulation method for quantum circuits based on decomposing unitary gates into a sum of stabilizer projectors. By only decomposing the non-Clifford gates, we take advantage of the Gottesman-Knill theorem and build a bridge between stabilizer-based simulation and Feynman-path type simulation. We give two variants of this method: stabilizer-based path integral recursion (SPIR) and stabilizer projector contraction (SPC). We analyze the advantages and disadvantages of our method compared to the Bravyi-Gosset algorithm and recursive Feynman path-integral algorithms. We construct a parametrized circuit ensemble and identify the parameter regime in this ensemble where our method offers superior performance. We also estimate the time cost for simulating quantum supremacy experiments with our method.

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“…As tangible quantum computers are now on the cusp of practical use [4][5][6][7][8], there is a growing requirement for formal methodologies regarding their verification to be developed. This requirement is complicated by the fact that there are no quantum computers available that could be used as a reference, it is therefore natural to endeavour to employ classical computers as the solution [6,9,10]. However, the use of classical computers quickly becomes infeasible as the size of the quantum computer increases.…”

Section: Introductionmentioning

confidence: 99%

Invariant subspaces of two-qubit quantum gates and their application in the verification of quantum computers

Yordanov¹,

Jacob²,

Ferrus³

et al. 2020

Preprint

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We investigate the groups generated by the sets of CP , CN OT and SW AP α (power-of-SWAP) quantum gate operations acting on n qubits. Isomorphisms to standard groups are found, and using techniques from representation theory, we are able to determine the invariant subspaces of the n−qubit Hilbert space under the action of each group. For the CP operation, we find isomorphism to the direct product of n(n − 1)/2 cyclic groups of order 2, and determine 2 n 1-dimensional invariant subspaces corresponding to the computational state-vectors. For the CN OT operation, we find isomorphism to the general linear group of an n-dimensional space over a field of 2 elements, GL(n, 2), and determine two 1-dimensional invariant subspaces and one (2 n − 2)-dimensional invariant subspace. For the SW AP α operation we determine a complex structure of invariant subspaces with varying dimensions and occurrences and present a recursive procedure to construct them. As an example of an application for our work, we suggest that these invariant subspaces can be used to construct simple formal verification procedures to assess the operation of quantum computers of arbitrary size.

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Simulating quantum computers with probabilistic methods

Cited by 4 publications

References 0 publications

Classical algorithms for Forrelation

Classical algorithms for Forrelation

Feynman-path type simulation using stabilizer projector decomposition of unitaries

Invariant subspaces of two-qubit quantum gates and their application in the verification of quantum computers

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