Measurement-Based Variational Quantum Eigensolver

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Measurement-Based Variational Quantum Eigensolver

R. R. Ferguson, L. Dellantonio, A. Al Balushi, K. Jansen, W. Dür, and C. A. Muschik

Phys. Rev. Lett. 126, 220501 – Published 1 June 2021

See Research News: Upgrading a Hybrid Computing Algorithm

Abstract

Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entangled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit- to measurement-based schemes. Both schemes offer problem-specific advantages in terms of the required resources and coherence times.

Received 5 November 2020
Revised 8 March 2021
Accepted 9 March 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.220501

Physics Subject Headings (PhySH)

Measurement-based quantum computing Quantum algorithms Quantum computation Quantum simulation

Quantum Information, Science & Technology

Research News

Upgrading a Hybrid Computing Algorithm

Published 8 June 2021

Researchers outline a protocol for performing a popular quantum-classical machine-learning algorithm with a so-called measurement-based quantum computer, which could allow for more resource-efficient calculations.

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Authors & Affiliations

R. R. Ferguson ^1,*, L. Dellantonio ^1,*, A. Al Balushi¹, K. Jansen², W. Dür ³, and C. A. Muschik ^1,4

¹Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo N2L 3G1, Canada
²NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany
³Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21a, 6020 Innsbruck, Austria
⁴Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

^*R. R. F. and L. D. contributed equally to this work.

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Vol. 126, Iss. 22 — 4 June 2021

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Images

Figure 1
MB-VQE schemes. (a) Variation of a problem-specific ansatz state by “edge decoration.” An ansatz graph state starts the MB-VQE in a suitable corner of Hilbert space (choice of green island). Next, a classical algorithm explores the neighborhood (runner on black arrow). The variational optimization exploits a custom state that is obtained by decorating the edges of the ansatz state with auxiliary qubits (orange circles). Their measurement in rotated bases $R (θ)$ with variational parameters $θ$ transforms $| ψ_{a} ⟩$ into the output state $| ψ_{out} ⟩$ . (b) Direct translation of a VQE circuit into a MB-VQE. Left: circuit consisting of Clifford gates (black) and single-qubit parametric gates (“knobs”). Right: corresponding MB-VQE, where the Clifford part of the circuit has been performed beforehand. The custom state consists of output (white circles) and auxiliary (orange circles) qubits only; the latter are measured in rotated bases and are related to the knobs in the circuit.
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Figure 2
Perturbed toric code. (a) Edge modification resource for the MB-VQE [see Fig. 1]. Four auxiliary qubits (orange circles), labeled $(m, n)_{i}$ ( $i = 1, \dots, 4$ ), are added to two connected output qubits $m$ and $n$ (white circles). (b) Graph state representation of the ansatz state $| 0, 0 ⟩_{L}$ . Additional Hadamard gates are applied to qubits with dashed lines. (c) Relative difference between the MB-VQE result and the true ground state energy vs the perturbation strength. We let $λ_{n}$ in Eq. (1) be equal on all qubits (solid blue line) or sampled from a normal distribution $P_{G}$ of average $λ$ and variance $0.1 λ$ (red squares). Green triangles describe a perturbation acting strongly on $λ_{1}$ and weakly on the other qubits. Dotted and dashed lines are computed with respect to $| 0, 0 ⟩_{L}$ (ansatz state) and $| 1 ⟩^{\otimes 2 N_{x} N_{y}}$ (ground state of ${\hat{H}}_{p}$ ) for $λ_{n} = λ \forall n$ .
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Figure 3
Schwinger model. (a) Ansatz state and VQE circuit for $S = 4$ qubits and $K$ layers. Each layer consists of $C X$ gates and local rotations (orange) parametrized by angles $θ_{ν, i}^{n}$ (with rotation axis $ν = x$ , $z$ ; $i = 1, \dots, 4$ ). (b) MB-VQE custom state for $K$ layers. White circles are output qubits. Auxiliary qubits (orange) are measured in a rotated bases $R (θ)$ . (c) The order parameter $⟨ \hat{O} ⟩$ vs the fermion mass $μ$ . The dashed line and dots represent exact diagonalization (ED) and (MB-)VQE results, respectively, with the number of layers $K$ indicated in the legend. The inset shows the infidelity $1 - F$ . (d) Relative energy difference $Δ E / E$ between (MB-)VQE results and ED for $μ = - 0.7$ vs the number of iterations in the optimization procedure. The variational parameters are initialized at zero, and $J = ω = 1$ in Eq. (2).
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