scholarly journals Noise-induced barren plateaus in variational quantum algorithms

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Samson Wang ◽  
Enrico Fontana ◽  
M. Cerezo ◽  
Kunal Sharma ◽  
Akira Sone ◽  
...  
Keyword(s):  
Quantum Algorithms ◽  
Random Parameter ◽  
Noise Model ◽  
Coupled Cluster ◽  
Natural Question ◽  
Intermediate Scale ◽  
Fundamental Limitations ◽  
Special Cases

AbstractVariational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n. These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model.

The Projective Antecedent of the Three Reflection Theorem

1991 ◽  
Vol 34 (2) ◽  
pp. 265-274
Author(s):  
F. A. Sherk
Keyword(s):  
Projective Geometry ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Metric Geometry ◽  
Natural Question ◽  
Complete Answer ◽  
Special Cases ◽  
Reflection Theorem ◽  
Necessary And Sufficient

AbstractA complete answer is given to the question: Under what circumstances is the product of three harmonic homologies in PG(2, F) again a harmonic homology ? This is the natural question to ask in seeking a generalization to projective geometry of the Three Reflection Theorem of metric geometry. It is found that apart from two familiar special cases, and with one curious exception, the necessary and sufficient conditions on the harmonic homologies produce exactly the Three Reflection Theorem.

scholarly journals Quantum algorithm for multivariate polynomial interpolation

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170480 ◽  
Author(s):  
Jianxin Chen ◽  
Andrew M. Childs ◽  
Shih-Han Hung
Keyword(s):  
Open Problem ◽  
Polynomial Interpolation ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Large Field ◽  
Multivariate Polynomial ◽  
Multivariate Polynomial Interpolation ◽  
Special Cases ◽  
Speed Up

How many quantum queries are required to determine the coefficients of a degree- d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ⌈ ( 1 / ( n + 1 ) ) ( n + d d ) ⌉ except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ queries suffice to achieve probability approaching 1 for large field order q . The classical query complexity of this problem is ( n + d d ) , so our result provides a speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and F q , respectively. Thus, we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of F q , we conjecture that 2 k C queries also suffice to achieve probability approaching 1 for large field order q , although we leave this as an open problem.

EFFICIENT QUANTUM ALGORITHMS FOR SOME INSTANCES OF THE NON-ABELIAN HIDDEN SUBGROUP PROBLEM

2003 ◽  
Vol 14 (05) ◽  
pp. 723-739 ◽  
Author(s):  
GÁBOR IVANYOS ◽  
FRÉDÉRIC MAGNIEZ ◽  
MIKLOS SANTHA
Keyword(s):  
Commutator Subgroup ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Solvable Groups ◽  
Normal Subgroups ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem ◽  
Small Index ◽  
Hidden Subgroups

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

scholarly journals An Efficient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups

2017 ◽  
Vol 18 (2) ◽  
pp. 0215 ◽  
Author(s):  
Demerson Nunes Gonçalves ◽  
Tharso D Fernandes ◽  
C M M Cosme
Keyword(s):  
Positive Integer ◽  
Quantum Computation ◽  
Prime Number ◽  
Abelian Groups ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Prime Factorization ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem

The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups $\Z_{N}\rtimes\Z_{q^s}$ where $N$ is an integer with a special prime factorization, $q$ prime number and $s$ any positive integer.

scholarly journals Fisher Information in Noisy Intermediate-Scale Quantum Applications

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 539
Author(s):  
Johannes Jakob Meyer
Keyword(s):  
Fisher Information ◽  
Quantum Algorithms ◽  
Quantum Fisher Information ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Quantum Computers ◽  
Quantum Devices ◽  
Intermediate Scale ◽  
Quantum Sensing ◽  
Near Term

The recent advent of noisy intermediate-scale quantum devices, especially near-term quantum computers, has sparked extensive research efforts concerned with their possible applications. At the forefront of the considered approaches are variational methods that use parametrized quantum circuits. The classical and quantum Fisher information are firmly rooted in the field of quantum sensing and have proven to be versatile tools to study such parametrized quantum systems. Their utility in the study of other applications of noisy intermediate-scale quantum devices, however, has only been discovered recently. Hoping to stimulate more such applications, this article aims to further popularize classical and quantum Fisher information as useful tools for near-term applications beyond quantum sensing. We start with a tutorial that builds an intuitive understanding of classical and quantum Fisher information and outlines how both quantities can be calculated on near-term devices. We also elucidate their relationship and how they are influenced by noise processes. Next, we give an overview of the core results of the quantum sensing literature and proceed to a comprehensive review of recent applications in variational quantum algorithms and quantum machine learning.

scholarly journals Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: Can quantum algorithms outperform their classical equivalents?

2020 ◽  
Vol 152 (12) ◽  
pp. 124107 ◽  
Author(s):  
Igor O. Sokolov ◽  
Panagiotis Kl. Barkoutsos ◽  
Pauline J. Ollitrault ◽  
Donny Greenberg ◽  
Julia Rice ◽  
...  
Keyword(s):  
Quantum Algorithms ◽  
Coupled Cluster ◽  
Strongly Correlated ◽  
Coupled Cluster Methods ◽  
Cluster Methods

scholarly journals Test of the unitary coupled-cluster variational quantum eigensolver for a simple strongly correlated condensed-matter system

2020 ◽  
Vol 34 (19n20) ◽  
pp. 2040049
Author(s):  
Luogen Xu ◽  
J. T. Lee ◽  
J. K. Freericks
Keyword(s):  
Condensed Matter ◽  
Quantum Circuit ◽  
Coupled Cluster ◽  
Quantum Computers ◽  
Strongly Correlated ◽  
Intermediate Scale ◽  
Condensed Matter System ◽  
Matter System ◽  
Nontrivial Behavior ◽  
Half Filling

The variational quantum eigensolver has been proposed as a low-depth quantum circuit that can be employed to examine strongly correlated systems on today’s noisy intermediate-scale quantum computers. We examine details associated with the factorized form of the unitary coupled-cluster variant of this algorithm. We apply it to a simple strongly correlated condensed-matter system with nontrivial behavior — the four-site Hubbard model at half-filling. This work show some of the subtle issues one needs to take into account when applying this algorithm in practice, especially to condensed-matter systems.

scholarly journals A quantum extension of SVM-perf for training nonlinear SVMs in almost linear time

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 342
Author(s):  
Jonathan Allcock ◽  
Chang-Yu Hsieh
Keyword(s):  
Input Data ◽  
Linear Time ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Large Data ◽  
Low Rank ◽  
Support Vector ◽  
Svm Model ◽  
Special Cases ◽  
Soft Margin

We propose a quantum algorithm for training nonlinear support vector machines (SVM) for feature space learning where classical input data is encoded in the amplitudes of quantum states. Based on the classical SVM-perf algorithm of Joachims \cite{joachims2006training}, our algorithm has a running time which scales linearly in the number of training examples m (up to polylogarithmic factors) and applies to the standard soft-margin ℓ1-SVM model. In contrast, while classical SVM-perf has demonstrated impressive performance on both linear and nonlinear SVMs, its efficiency is guaranteed only in certain cases: it achieves linear m scaling only for linear SVMs, where classification is performed in the original input data space, or for the special cases of low-rank or shift-invariant kernels. Similarly, previously proposed quantum algorithms either have super-linear scaling in m, or else apply to different SVM models such as the hard-margin or least squares ℓ2-SVM which lack certain desirable properties of the soft-margin ℓ1-SVM model. We classically simulate our algorithm and give evidence that it can perform well in practice, and not only for asymptotically large data sets.

scholarly journals Localized Quantum Chemistry on Quantum Computers

2021 ◽  
Author(s):  
Matthew Otten ◽  
Matthew Hermes ◽  
Riddhish Pandharkar ◽  
Yuri Alexeev ◽  
Stephen Gray ◽  
...  
Keyword(s):  
Quantum Chemistry ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
System Size ◽  
Coupled Cluster ◽  
Quantum Computers ◽  
Active Space ◽  
Polynomial Reduction ◽  
Quantum Chemistry Calculations ◽  
Self Consistent Field

Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today’s quantum devices. Here we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground state energy. Our algorithm, termed “local active space unitary coupled cluster” (LAS-UCC), scales linearly with system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2)2 into two H2 molecules and by breaking the two double bonds in trans-butadiene and resources estimates are provided for linear chains of up to 20 H2 molecules.

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