A diagrammatic approach to variational quantum ansatz construction

Y. Herasymenko and T.E. O'Brien

Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

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Abstract

Variational quantum eigensolvers (VQEs) are a promising class of quantum algorithms for preparing approximate ground states in near-term quantum devices. Minimizing the error in such an approximation requires designing ansatzes using physical considerations that target the studied system. One such consideration is size-extensivity, meaning that the ground state quantum correlations are to be compactly represented in the ansatz. On digital quantum computers, however, the size-extensive ansatzes usually require expansion via Trotter-Suzuki methods. These introduce additional costs and errors to the approximation. In this work, we present a diagrammatic scheme for the digital VQE ansatzes, which is size-extensive but does not rely on Trotterization. We start by designing a family of digital ansatzes that explore the entire Hilbert space with the minimum number of free parameters. We then demonstrate how one may compress an arbitrary digital ansatz, by enforcing symmetry constraints of the target system, or by using them as parent ansatzes for a hierarchy of increasingly long but increasingly accurate sub-ansatzes. We apply a perturbative analysis and develop a diagrammatic formalism that ensures the size-extensivity of generated hierarchies. We test our methods on a short spin chain, finding good convergence to the ground state in the paramagnetic and the ferromagnetic phase of the transverse-field Ising model.

In the near-term, some of the most promising applications for quantum computers involve tuning a "templated" quantum protocol – a variational ansatz – to simulate the low-energy states of complex quantum systems. For high performance, such ansatzes should fit the physics of the problem. Many popular ansatzes, like unitary coupled-cluster, aim to ensure this by compactly representing system’s correlations, as is rigorously defined by the principle of size-extensivity. However, digitizing these into standard operations typically requires the use of the inexact Trotter expansion.

In this work, we resolve this conflict with a framework for size-extensive ansatzes, which is fundamentally digital and therefore involves no Trotter errors. We construct a family of ansatzes that provably cover the entire space of quantum register states with a minimal number of parameters. We show how to compress such parent ansatzes into practical child ansatzes that target specific systems. For that, we use symmetries and perturbation theory (for which we develop a convenient diagrammatic approach). We find good convergence of our method for the quantum Ising chain away from the quantum phase transition, in agreement with our theoretical expectations.

We expect this work to be of use both in near-term quantum hardware implementations of variational algorithms (where low circuit depths are critical) and in large-scale applications where variational ansatzes can only explore minuscule regions of the large N-qubit Hilbert space.

► BibTeX data

► References

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[22] J. Kirkwood and L. Thomas, Expansions and phase transitions for the ground state of quantum Ising lattice systems, Commun. Math. Phys. 88, 569 (1983).
https:/​/​doi.org/​10.1007/​BF01211959

[23] S. Bravyi, D. DiVincenzo, and D. Loss, Polynomial-time algorithm for simulation of weakly interacting quantum spin systems , Commun. Math. Phys. 284, 481 (2008).
https:/​/​doi.org/​10.1007/​s00220-008-0574-6

[24] D. Wecker, M. B. Hastings, and M. Troyer, Progress towards practical quantum variational algorithms, Phys. Rev. A 92, 042303 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.042303

[1] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[2] D. Litinski, Magic State Distillation: Not as Costly as You Think, Quantum 3, 205 (2019).
https:/​/​doi.org/​10.22331/​q-2019-12-02-205

[3] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J.Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Comm. 5, 4213 (2014).
https:/​/​doi.org/​10.1038/​ncomms5213

[4] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New J. Phys. 18, 023023 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023

[5] J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik, Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz , Quantum Sci. Technol. 4, 014008 (2018).
https:/​/​doi.org/​10.1088/​2058-9565/​aad3e4

[6] J. McClean, S. Boixo, V. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes , Nat. Comm. 9, 4812 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-07090-4

[7] P.-L. Dallaire-Demers, J. Romero, L. Veis, S. Sim, and A. Aspuru-Guzik, Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer, Quantum Sci. Technol. 4, 045005 (2019).
https:/​/​doi.org/​10.1088/​2058-9565/​ab3951

[8] E. Farhi, J. Goldstone, and S. Gutmann, A Quantum Approximate Optimization Algorithm, arXiv:1411.4028.
arXiv:1411.4028

[9] S. Lloyd, Quantum approximate optimization is computationally universal , arXiv:1812.11075.
arXiv:1812.11075

[10] K. A. Brueckner, Many-Body Problem for Strongly Interacting Particles. II. Linked Cluster Expansion, Phys. Rev. 100, 36 (1955).
https:/​/​doi.org/​10.1103/​PhysRev.100.36

[11] H. F. Trotter, On the Product of Semi-Groups of Operators, Proc. Am. Math. Soc. 10, 545 (1959).
https:/​/​doi.org/​10.1090/​S0002-9939-1959-0108732-6

[12] M. Suzuki, General theory of fractal path integrals with applications to many‐body theories and statistical physics, J. Math. Phys. 32 (1991).
https:/​/​doi.org/​10.1063/​1.529425

[13] J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Simulation of electronic structure Hamiltonians using quantum computers , Mol. Phys. 109, 735 (2011).
https:/​/​doi.org/​10.1080/​00268976.2011.552441

[14] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M.Chow, and J. M. Gambetta, Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets , Nature 549, 242 (2017).
https:/​/​doi.org/​10.1038/​nature23879

[15] R. Sagastizabal, X. Bonet-Monroig, M. Singh, M. Rol, C. Bultink, X. Fu, C. Price, V. Ostroukh, N. Muthusubramanian, A. Bruno, M. Beekman, N. Haider, T. O’Brien, and L. DiCarlo, Error Mitigation by Symmetry Verification on a Variational Quantum Eigensolver, Phys. Rev. A 100, 010302 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.010302

[16] G. Guerreschi and M. Smelyanskiy, Practical optimization for hybrid quantum-classical algorithms, arXiv:1701.01450.
arXiv:1701.01450

[17] O. Higgott, D. Wang, and S. Brierley, Variational Quantum Computation of Excited States, Quantum 3, 156 (2019).
https:/​/​doi.org/​10.22331/​q-2019-07-01-156

[18] S. Endo, T. Jones, S. McArdle, X. Yuan, and S. Benjamin, Variational quantum algorithms for discovering Hamiltonian spectra, Phys. Rev. A 99, 062304 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.062304

[19] K. M. Nakanishi, K. Fujii, and S. Todo, Sequential minimal optimization for quantum-classical hybrid algorithms, Phys. Rev. Research 2, 043158 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.043158

[20] D. Gottesman, Stabilizer Codes and Quantum Error Correction, PhD Dissertation, California Institute of Technology (1997).
arXiv:quant-ph/9705052

[21] B.T. Gard, L. Zhu, G.S. Barron, N.J. Mayhall, S.E. Economou, and E. Barnes, Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm, NPJ Quantum Inf. 6, 10 (2020).
https:/​/​doi.org/​10.1038/​s41534-019-0240-1

[22] J. Kirkwood and L. Thomas, Expansions and phase transitions for the ground state of quantum Ising lattice systems, Commun. Math. Phys. 88, 569 (1983).
https:/​/​doi.org/​10.1007/​BF01211959

[23] S. Bravyi, D. DiVincenzo, and D. Loss, Polynomial-time algorithm for simulation of weakly interacting quantum spin systems , Commun. Math. Phys. 284, 481 (2008).
https:/​/​doi.org/​10.1007/​s00220-008-0574-6

[24] D. Wecker, M. B. Hastings, and M. Troyer, Progress towards practical quantum variational algorithms, Phys. Rev. A 92, 042303 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.042303

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[1] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits", Physical Review A 103 5, 052416 (2021).

[2] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian Operator Approximation for Energy Measurement and Ground-State Preparation", PRX Quantum 2 3, 030318 (2021).

[3] M. E. S. Morales, J. D. Biamonte, and Z. Zimborás, "On the universality of the quantum approximate optimization algorithm", Quantum Information Processing 19 9, 291 (2020).

[4] Thomas E. O'Brien, Bruno Senjean, Ramiro Sagastizabal, Xavier Bonet-Monroig, Alicja Dutkiewicz, Francesco Buda, Leonardo DiCarlo, and Lucas Visscher, "Calculating energy derivatives for quantum chemistry on a quantum computer", npj Quantum Information 5, 113 (2019).

[5] D. Chivilikhin, A. Samarin, V. Ulyantsev, I. Iorsh, A. R. Oganov, and O. Kyriienko, "MoG-VQE: Multiobjective genetic variational quantum eigensolver", arXiv:2007.04424.

[6] Oleksandr Kyriienko, "Quantum inverse iteration algorithm for programmable quantum simulators", npj Quantum Information 6, 7 (2020).

[7] Tatiana A. Bespalova and Oleksandr Kyriienko, "Quantum simulation and ground state preparation for the honeycomb Kitaev model", arXiv:2109.13883.

[8] Pejman Jouzdani and Stefan Bringuier, "Hybrid Quantum-Classical Eigensolver Without Variation or Parametric Gates", arXiv:2008.11347.

[9] Joseph C. Aulicino, Trevor Keen, and Bo Peng, "State preparation and evolution in quantum computing: a perspective from Hamiltonian moments", arXiv:2109.12790.

The above citations are from SAO/NASA ADS (last updated successfully 2022-07-05 17:09:33). The list may be incomplete as not all publishers provide suitable and complete citation data.

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