Noise-induced barren plateaus in variational quantum algorithms

Variational 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.

[1]  Ying Li,et al.  Variational algorithms for linear algebra. , 2019, Science bulletin.

[2]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[3]  L. Banchi,et al.  Noise-resilient variational hybrid quantum-classical optimization , 2019, Physical Review A.

[4]  Ryan Babbush,et al.  Barren plateaus in quantum neural network training landscapes , 2018, Nature Communications.

[5]  Masoud Mohseni,et al.  Observation of separated dynamics of charge and spin in the Fermi-Hubbard model , 2020, 2010.07965.

[6]  Harper R. Grimsley,et al.  An adaptive variational algorithm for exact molecular simulations on a quantum computer , 2018, Nature Communications.

[7]  Ryan Babbush,et al.  The theory of variational hybrid quantum-classical algorithms , 2015, 1509.04279.

[8]  V. Ulyantsev,et al.  MoG-VQE: Multiobjective genetic variational quantum eigensolver , 2020, 2007.04424.

[9]  Maria Schuld,et al.  The quest for a Quantum Neural Network , 2014, Quantum Information Processing.

[10]  Rupak Biswas,et al.  From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz , 2017, Algorithms.

[11]  Patrick J. Coles,et al.  Variational quantum state diagonalization , 2018, npj Quantum Information.

[12]  Alán Aspuru-Guzik,et al.  Quantum computational chemistry , 2018, Reviews of Modern Physics.

[13]  Stuart Hadfield,et al.  The Quantum Approximation Optimization Algorithm for MaxCut: A Fermionic View , 2017, 1706.02998.

[14]  Peter D. Johnson,et al.  QVECTOR: an algorithm for device-tailored quantum error correction , 2017, 1711.02249.

[15]  Patrick J. Coles,et al.  Variational consistent histories as a hybrid algorithm for quantum foundations , 2018, Nature Communications.

[16]  M. Cerezo,et al.  A semi-agnostic ansatz with variable structure for quantum machine learning , 2021, arXiv.org.

[17]  Markus Brink,et al.  Demonstration of quantum volume 64 on a superconducting quantum computing system , 2020, Quantum Science and Technology.

[18]  M. B. Hastings,et al.  Classical and quantum bounded depth approximation algorithms , 2019, Quantum Inf. Comput..

[19]  P. Ginsparg,et al.  Experimental error mitigation using linear rescaling for variational quantum eigensolving with up to 20 qubits , 2021, Quantum Science and Technology.

[20]  Sukin Sim,et al.  Noisy intermediate-scale quantum (NISQ) algorithms , 2021, Reviews of Modern Physics.

[21]  Xiao Yuan,et al.  Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation , 2020, Journal of the Physical Society of Japan.

[22]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[23]  Alán Aspuru-Guzik,et al.  A variational eigenvalue solver on a photonic quantum processor , 2013, Nature Communications.

[24]  Patrick J. Coles,et al.  Variational Quantum State Eigensolver , 2020, 2004.01372.

[25]  Jacob biamonte,et al.  Quantum machine learning , 2016, Nature.

[26]  Masoud Mohseni,et al.  Layerwise learning for quantum neural networks , 2020, Quantum Machine Intelligence.

[27]  Jakob S. Kottmann,et al.  Mutual information-assisted adaptive variational quantum eigensolver , 2020, Quantum Science and Technology.

[28]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[29]  Akira Sone,et al.  Cost-Function-Dependent Barren Plateaus in Shallow Quantum Neural Networks , 2020, ArXiv.

[30]  Jack Hidary,et al.  Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm , 2019, ArXiv.

[31]  Alán Aspuru-Guzik,et al.  Quantum Chemistry in the Age of Quantum Computing. , 2018, Chemical reviews.

[32]  F. Petruccione,et al.  An introduction to quantum machine learning , 2014, Contemporary Physics.

[33]  K. Audenaert,et al.  Impressions of convexity: An illustration for commutator bounds , 2010, 1004.2700.

[34]  Bryan O'Gorman,et al.  Generalized swap networks for near-term quantum computing , 2019, ArXiv.

[35]  Peter Maunz,et al.  Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography , 2016, Nature Communications.

[36]  B'alint Koczor,et al.  Quantum Analytic Descent , 2020 .

[37]  Tyson Jones,et al.  Quantum compilation and circuit optimisation via energy dissipation , 2018 .

[38]  Edward Grant,et al.  An initialization strategy for addressing barren plateaus in parametrized quantum circuits , 2019, Quantum.

[39]  Alán Aspuru-Guzik,et al.  Variational Quantum Factoring , 2018, QTOP@NetSys.

[40]  Nooijen.  Can the eigenstates of a many-body hamiltonian Be represented exactly using a general two-body cluster expansion? , 2000, Physical review letters.

[41]  Robert Koenig,et al.  Obstacles to State Preparation and Variational Optimization from Symmetry Protection. , 2019, 1910.08980.

[42]  Daniel Stilck França,et al.  Relative Entropy Convergence for Depolarizing Channels , 2015, 1508.07021.

[43]  Gavin E. Crooks,et al.  Performance of the Quantum Approximate Optimization Algorithm on the Maximum Cut Problem , 2018, 1811.08419.

[44]  Ying Li,et al.  Theory of variational quantum simulation , 2018, Quantum.

[45]  Patrick J. Coles,et al.  Machine Learning of Noise-Resilient Quantum Circuits , 2020, PRX Quantum.

[46]  M. Cerezo,et al.  Effect of barren plateaus on gradient-free optimization , 2020, Quantum.

[47]  B. Nachman,et al.  Zero-noise extrapolation for quantum-gate error mitigation with identity insertions , 2020, Physical Review A.

[48]  Marco Pistoia,et al.  A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver , 2019 .

[49]  A. Montanaro,et al.  Error mitigation by training with fermionic linear optics , 2021, 2102.02120.

[50]  Tobias J. Osborne,et al.  Training deep quantum neural networks , 2020, Nature Communications.

[51]  Masoud Mohseni,et al.  Learning to learn with quantum neural networks via classical neural networks , 2019, ArXiv.

[52]  John Preskill,et al.  Quantum Computing in the NISQ era and beyond , 2018, Quantum.

[53]  Patrick J. Coles,et al.  Cost function dependent barren plateaus in shallow parametrized quantum circuits , 2021, Nature Communications.

[54]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[55]  Patrick J. Coles,et al.  Operator Sampling for Shot-frugal Optimization in Variational Algorithms , 2020, 2004.06252.

[56]  B'alint Koczor,et al.  Variational-state quantum metrology , 2019, New Journal of Physics.

[57]  On contraction coefficients, partial orders and approximation of capacities for quantum channels , 2020, ArXiv.

[58]  Patrick J. Coles,et al.  Variational fast forwarding for quantum simulation beyond the coherence time , 2019, npj Quantum Information.

[59]  Cheng Xue,et al.  Effects of Quantum Noise on Quantum Approximate Optimization Algorithm , 2019 .

[60]  Patrick J. Coles,et al.  Variational Quantum Fidelity Estimation , 2019, Quantum.

[61]  Patrick J. Coles,et al.  An Adaptive Optimizer for Measurement-Frugal Variational Algorithms , 2019 .

[62]  Patrick J. Coles,et al.  Higher order derivatives of quantum neural networks with barren plateaus , 2020, 2008.07454.

[63]  Stefan Woerner,et al.  The power of quantum neural networks , 2020, Nature Computational Science.

[64]  Efficient Mitigation of Depolarizing Errors in Quantum Simulations , 2021, 2101.01690.

[65]  Jens Eisert,et al.  A variational toolbox for quantum multi-parameter estimation , 2020, npj Quantum Information.

[66]  R. Bartlett,et al.  Coupled-cluster theory in quantum chemistry , 2007 .

[67]  Robert König,et al.  Quantum entropy and its use , 2017 .

[68]  Xiao Yuan,et al.  Variational quantum algorithms for discovering Hamiltonian spectra , 2018, Physical Review A.

[69]  Matthew B. Hastings,et al.  Hybrid quantum-classical approach to correlated materials , 2015, 1510.03859.

[70]  A V Uvarov,et al.  On barren plateaus and cost function locality in variational quantum algorithms , 2021, Journal of Physics A: Mathematical and Theoretical.

[71]  B. Baumgartner.  An inequality for the trace of matrix products, using absolute values , 2011, 1106.6189.

[72]  V. Akshay,et al.  Reachability Deficits in Quantum Approximate Optimization , 2019, Physical review letters.

[73]  A. Harrow,et al.  Quantum Supremacy through the Quantum Approximate Optimization Algorithm , 2016, 1602.07674.

[74]  Kunal Sharma,et al.  Trainability of Dissipative Perceptron-Based Quantum Neural Networks , 2020, ArXiv.

[75]  S. Ronen,et al.  Can the eigenstates of a many-body Hamiltonian be represented exactly using a general two-body cluster expansion? , 2003, Physical review letters.

[76]  Ryan LaRose,et al.  Quantum-assisted quantum compiling , 2018, Quantum.

[77]  M. Hastings,et al.  Progress towards practical quantum variational algorithms , 2015, 1507.08969.

[78]  Patrick J. Coles,et al.  Large gradients via correlation in random parameterized quantum circuits , 2020, Quantum Science and Technology.

[79]  S. Yelin,et al.  Entanglement devised barren plateau mitigation , 2020, Physical Review Research.

[80]  E. Knill,et al.  Quantum algorithms for fermionic simulations , 2000, cond-mat/0012334.

[81]  Yvette de Sereville,et al.  Exploring entanglement and optimization within the Hamiltonian Variational Ansatz , 2020, PRX Quantum.

[82]  Patrick J. Coles,et al.  Variational Quantum Linear Solver: A Hybrid Algorithm for Linear Systems , 2019, 1909.05820.

[83]  Stuart Hadfield,et al.  Characterizing local noise in QAOA circuits , 2020, IOP SciNotes.

[84]  Harper R. Grimsley,et al.  qubit-ADAPT-VQE: An adaptive algorithm for constructing hardware-efficient ansatze on a quantum processor , 2019, 1911.10205.

[85]  Nathan Wiebe,et al.  Entanglement Induced Barren Plateaus , 2020, PRX Quantum.

[86]  Patrick J. Coles,et al.  Impact of Barren Plateaus on the Hessian and Higher Order Derivatives. , 2020 .

[87]  Kunal Sharma,et al.  Noise resilience of variational quantum compiling , 2019, New Journal of Physics.

[88]  W. W. Ho,et al.  Efficient variational simulation of non-trivial quantum states , 2018, SciPost Physics.

[89]  Ying Li,et al.  Efficient Variational Quantum Simulator Incorporating Active Error Minimization , 2016, 1611.09301.

[90]  Tyler Y Takeshita,et al.  Hartree-Fock on a superconducting qubit quantum computer , 2020, Science.

[91]  E. Rieffel,et al.  Near-optimal quantum circuit for Grover's unstructured search using a transverse field , 2017, 1702.02577.

[92]  Patrick J. Coles,et al.  Error mitigation with Clifford quantum-circuit data , 2020, Quantum.

[93]  Kunal Sharma,et al.  Connecting ansatz expressibility to gradient magnitudes and barren plateaus , 2021, ArXiv.

[94]  E. Farhi,et al.  A Quantum Approximate Optimization Algorithm , 2014, 1411.4028.

[95]  Ken M. Nakanishi,et al.  Subspace variational quantum simulator , 2019, Physical Review Research.

[96]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[97]  L. Landau,et al.  Fermionic quantum computation , 2000 .

[98]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[99]  J. Gambetta,et al.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets , 2017, Nature.

[100]  K. B. Whaley,et al.  Generalized Unitary Coupled Cluster Wave functions for Quantum Computation. , 2018, Journal of chemical theory and computation.

[101]  R. Blume-Kohout,et al.  Probing quantum processor performance with pyGSTi , 2020, Quantum Science and Technology.

[102]  Patrick J. Coles,et al.  Learning the quantum algorithm for state overlap , 2018, New Journal of Physics.

[103]  J. McClean,et al.  Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz , 2017, Quantum Science and Technology.

[104]  Tao Huang,et al.  Quantum circuit architecture search: error mitigation and trainability enhancement for variational quantum solvers , 2020, ArXiv.

[105]  A. Shaw.  Classical-Quantum Noise Mitigation for NISQ Hardware , 2021, 2105.08701.

[106]  M. Cerezo,et al.  Variational quantum algorithms , 2020, Nature Reviews Physics.

[107]  A. Kitaev,et al.  Fermionic Quantum Computation , 2000, quant-ph/0003137.

[108]  D. Bacon,et al.  Quantum approximate optimization of non-planar graph problems on a planar superconducting processor , 2020, Nature Physics.