Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans{ä}tze

We develop a theoretical framework for $S_n$-equivariant convolutional quantum circuits with SU$(d)$-symmetry, building on and significantly generalizing Jordan's Permutational Quantum Computing (PQC) formalism based on Schur-Weyl duality connecting both SU$(d)$ and $S_n$ actions on qudits. In particular, we utilize the Okounkov-Vershik approach to prove Harrow's statement (Ph.D. Thesis 2005 p.160) on the equivalence between $\operatorname{SU}(d)$ and $S_n$ irrep bases and to establish the $S_n$-equivariant Convolutional Quantum Alternating Ans\"atze ($S_n$-CQA) using Young-Jucys-Murphy (YJM) elements. We prove that $S_n$-CQA is able to generate any unitary in any given $S_n$ irrep sector, which may serve as a universal model for a wide array of quantum machine learning problems with the presence of SU($d$) symmetry. Our method provides another way to prove the universality of Quantum Approximate Optimization Algorithm (QAOA) and verifies that 4-local SU($d$) symmetric unitaries are sufficient to build generic SU($d$) symmetric quantum circuits up to relative phase factors. We present numerical simulations to showcase the effectiveness of the ans\"atze to find the ground state energy of the $J_1$--$J_2$ antiferromagnetic Heisenberg model on the rectangular and Kagome lattices. Our work provides the first application of the celebrated Okounkov-Vershik's $S_n$ representation theory to quantum physics and machine learning, from which to propose quantum variational ans\"atze that strongly suggests to be classically intractable tailored towards a specific optimization problem.

[1]  Tobias Pfaff,et al.  Learning ground states of quantum Hamiltonians with graph networks , 2021, ArXiv.

[2]  陽平 藤.  Physics and Mathematics of Quantum Many-Body Systems , 2021 .

[3]  T. Vieijra,et al.  Many-body quantum states with exact conservation of non-Abelian and lattice symmetries through variational Monte Carlo , 2021, Physical Review B.

[4]  A. MacDonald,et al.  Group Convolutional Neural Networks Improve Quantum State Accuracy , 2021, 2104.05085.

[5]  Maria Schuld,et al.  Quantum machine learning models are kernel methods , 2021, 2101.11020.

[6]  John Preskill,et al.  Information-theoretic bounds on quantum advantage in machine learning , 2021, Physical review letters.

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

[8]  M. Schmitt,et al.  Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape , 2020, SciPost Physics.

[9]  Ö. Eğecioğlu,et al.  Young’s Seminormal Representation, Murphy Elements, and Content Evaluations , 2020 .

[10]  K. Temme,et al.  A rigorous and robust quantum speed-up in supervised machine learning , 2020, Nature Physics.

[11]  Wen-Yuan Liu,et al.  Gapless quantum spin liquid and global phase diagram of the spin-1/2 J1-J2 square antiferromagnetic Heisenberg model. , 2020, Science bulletin.

[12]  David W. Miller,et al.  Lorentz Group Equivariant Neural Network for Particle Physics , 2020, ICML.

[13]  Y. Nomura,et al.  Dirac-Type Nodal Spin Liquid Revealed by Refined Quantum Many-Body Solver Using Neural-Network Wave Function, Correlation Ratio, and Level Spectroscopy , 2020, Physical Review X.

[14]  H. Tasaki.  Physics and Mathematics of Quantum Many-Body Systems , 2020, Graduate Texts in Physics.

[15]  Erik H. Thiede,et al.  The general theory of permutation equivarant neural networks and higher order graph variational encoders , 2020, ArXiv.

[16]  I. Marvian.  Restrictions on realizable unitary operations imposed by symmetry and locality , 2020, Nature Physics.

[17]  A. Szab'o,et al.  Neural network wave functions and the sign problem , 2020, Physical Review Research.

[18]  Proceedings of the IEEE , 2020, IEEE Transactions on Automatic Control.

[19]  S. Yunoki,et al.  Symmetry-adapted variational quantum eigensolver , 2019, 1912.13146.

[20]  E. Farhi,et al.  The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size , 2019, Quantum.

[21]  Jacob Biamonte,et al.  On the universality of the quantum approximate optimization algorithm , 2019, Quantum Information Processing.

[22]  M. Katsnelson,et al.  Generalization properties of neural network approximations to frustrated magnet ground states , 2019, Nature Communications.

[23]  Guglielmo Mazzola,et al.  NetKet: A machine learning toolkit for many-body quantum systems , 2019, SoftwareX.

[24]  Kenny Choo,et al.  Two-dimensional frustrated J1−J2 model studied with neural network quantum states , 2019, Physical Review B.

[25]  Soonwon Choi,et al.  Quantum convolutional neural networks , 2018, Nature Physics.

[26]  Vojtěch Havlíček,et al.  Classical algorithm for quantum SU(2) Schur sampling , 2018, Physical Review A.

[27]  Yong-Sheng Zhang,et al.  Solving frustrated quantum many-particle models with convolutional neural networks , 2018, Physical Review B.

[28]  Zhen Lin,et al.  Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network , 2018, NeurIPS.

[29]  Kristan Temme,et al.  Supervised learning with quantum-enhanced feature spaces , 2018, Nature.

[30]  Hari Krovi,et al.  An efficient high dimensional quantum Schur transform , 2018, Quantum.

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

[32]  Risi Kondor,et al.  On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups , 2018, ICML.

[33]  Vojtěch Havlíček,et al.  Quantum Schur Sampling Circuits can be Strongly Simulated. , 2018, Physical review letters.

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

[35]  R. Pauncz.  The Symmetric Group in Quantum Chemistry , 2017 .

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

[37]  Frederick W. Strauch,et al.  A practical quantum algorithm for the Schur transform , 2017, Quantum Inf. Comput..

[38]  S. Lloyd,et al.  Quantum machine learning , 2016, Nature.

[39]  Alán Aspuru-Guzik,et al.  The theory of variational hybrid quantum-classical algorithms , 2015, 1509.04279.

[40]  J. Chalker,et al.  Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models , 2015, 1506.06798.

[41]  Geoffrey E. Hinton,et al.  Deep Learning , 2015, Nature.

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

[43]  Peter Wittek,et al.  Quantum Machine Learning: What Quantum Computing Means to Data Mining , 2014 .

[44]  S. Lloyd,et al.  Quantum principal component analysis , 2013, Nature Physics.

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

[46]  F. Verstraete,et al.  Tensor-product state approach to spin- 1 2 square J 1 − J 2 antiferromagnetic Heisenberg model: Evidence for deconfined quantum criticality , 2011, 1112.3331.

[47]  Karl-Hermann Neeb,et al.  Structure and Geometry of Lie Groups , 2011 .

[48]  L. Balents.  Spin liquids in frustrated magnets , 2010, Nature.

[49]  R. Goodman,et al.  Symmetry, Representations, and Invariants , 2009 .

[50]  Stephen P. Jordan,et al.  Permutational quantum computing , 2009, Quantum Inf. Comput..

[51]  P. Py.  On the Representation Theory of the Symmetric Groups , 2005 .

[52]  I. Marin.  L'algèbre de Lie des transpositions , 2005, math/0502119.

[53]  D. Klein,et al.  Symmetric‐group algebraic variational solutions for Heisenberg models at finite temperature , 1992 .

[54]  Berndt Müller,et al.  Quantum Mechanics: Symmetries , 1990 .

[55]  G. Murphy,et al.  A new construction of Young's seminormal representation of the symmetric groups , 1981 .

[56]  A.-A.A. Jucys,et al.  Symmetric polynomials and the center of the symmetric group ring , 1974 .

[57]  L. Biedenharn.  On the Representations of the Semisimple Lie Groups. I. The Explicit Construction of Invariants for the Unimodular Unitary Group in N Dimensions , 1963 .

[58]  E. Lieb,et al.  Ordering Energy Levels of Interacting Spin Systems , 1962 .

[59]  Masatake Kuranishi,et al.  On Everywhere Dense Imbedding of Free Groups in Lie Groups , 1951, Nagoya Mathematical Journal.

[60]  W. Heisenberg.  Zur Theorie des Ferromagnetismus , 1928 .

[61]  F. Murnaghan,et al.  LINEAR ALGEBRAIC GROUPS , 2005 .

[62]  S. Ormale,et al.  Quotients infinitésimaux du groupe de tresses , 2003 .

[63]  B. Sagan.  The Symmetric Group , 2001 .

[64]  Yoshua Bengio,et al.  Gradient-based learning applied to document recognition , 1998, Proc. IEEE.

[65]  A. L. Onishchik,et al.  Lie groups and algebraic groups , 1990 .

[66]  Alfred Young,et al.  The collected papers of Alfred Young 1873-1940 , 1977 .

[67]  L. Biedenharn,et al.  On the Representations of the Semisimple Lie Groups. II , 1963 .

[68]  E. Lieb,et al.  Physical Review Letters , 1958, Nature.

[69]  A. B. Lidiard,et al.  Antiferromagnetism , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[70]  Jstor.  Proceedings of the American Mathematical Society , 1950 .

[71]  E. B. Wilson,et al.  PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES. , 1916, Science.