A diagrammatic approach to variational quantum ansatz construction

Y. Herasymenko; T.E. O'Brien

doi:10.22331/q-2021-12-02-596

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

Published:	2021-12-02, volume 5, page 596
Eprint:	arXiv:1907.08157v3
Doi:	https://doi.org/10.22331/q-2021-12-02-596
Citation:	Quantum 5, 596 (2021).

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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.

Featured image: An example of our quantum combinatorial ansatz (QCA) circuit as applied to parameterize real states of $3$ qubits. For simplicity, we label each circuit element by the tensor factors of its generating Pauli operator on each qubit. E.g. the label XXY corresponds to the rotation $e^{i\theta_{XXY}XXY}$. For $N_q$ qubits, a general QCA contains $2(2^{N_q}-1)$ gates and is proven to cover the entire Hilbert space (which has exactly $2(2^{N_q}-1)$ degrees of freedom). This exponential complexity, however, is not suitable for a direct application in a variational algorithm. Instead, QCA is to be reduced to the desired number of gates via a diagrammatic approach outlined in our work. The size-extensivity of the resulting ansatz construction can be proven, with the underlying structure of QCA being the crucial ingredient.

Popular summary

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

@article{Herasymenko2021diagrammatic,
  doi = {10.22331/q-2021-12-02-596},
  url = {https://doi.org/10.22331/q-2021-12-02-596},
  title = {A diagrammatic approach to variational quantum ansatz construction},
  author = {Herasymenko, Y. and O'Brien, T.E.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {596},
  month = dec,
  year = {2021}
}

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[2] Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster, Antonio Anna Mele, Francesco Arzani, Alissa Wilms, and Jens Eisert, "Exploiting Symmetry in Variational Quantum Machine Learning", PRX Quantum 4 1, 010328 (2023).

[3] D. Chivilikhin, A. Samarin, V. Ulyantsev, I. Iorsh, A. R. Oganov, and O. Kyriienko, "MoG-VQE: Multiobjective genetic variational quantum eigensolver", arXiv:2007.04424, (2020).

[4] 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).

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

[6] 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).

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

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

[9] Tatiana A. Bespalova and Oleksandr Kyriienko, "Quantum simulation and ground state preparation for the honeycomb Kitaev model", arXiv:2109.13883, (2021).

[10] Chufan Lyu, Victor Montenegro, and Abolfazl Bayat, "Accelerated variational algorithms for digital quantum simulation of many-body ground states", Quantum 4, 324 (2020).

[11] Joseph C. Aulicino, Trevor Keen, and Bo Peng, "State preparation and evolution in quantum computing: a perspective from Hamiltonian moments", arXiv:2109.12790, (2021).

[12] Pejman Jouzdani and Stefan Bringuier, "Hybrid Quantum-Classical Eigensolver Without Variation or Parametric Gates", arXiv:2008.11347, (2020).

[13] Hui-Min Li, Zhi-Xi Wang, and Shao-Ming Fei, "Variational quantum algorithms for Poisson equations based on the decomposition of sparse Hamiltonians", arXiv:2309.12826, (2023).

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