Measurement of the Entanglement Spectrum of a Symmetry-Protected Topological State Using the IBM Quantum Computer

Kenny Choo, Curt W. von Keyserlingk, Nicolas Regnault, and Titus Neupert

Phys. Rev. Lett. 121, 086808 – Published 24 August 2018

Abstract

Entanglement properties are routinely used to characterize phases of quantum matter in theoretical computations. For example, the spectrum of the reduced density matrix, or so-called “entanglement spectrum”, has become a widely used diagnostic for universal topological properties of quantum phases. However, while being convenient to calculate theoretically, it is notoriously hard to measure in experiments. Here, we use the IBM quantum computer to make the first ever measurement of the entanglement spectrum of a symmetry-protected topological state. We are able to distinguish its entanglement spectrum from those we measure for trivial and long-range ordered states.

Received 7 May 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.086808

Physics Subject Headings (PhySH)

Quantum computation Symmetry protected topological states

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Kenny Choo¹, Curt W. von Keyserlingk², Nicolas Regnault³, and Titus Neupert¹

¹Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
²University of Birmingham, School of Physics and Astronomy, Birmingham B15 2TT, United Kingdom
³Laboratoire Pierre Aigrain, Ecole normale supérieure, PSL University, Sorbonne Université, Université Paris Diderot, Sorbonne Paris Cité, CNRS, 24 rue Lhomond, 75005 Paris, France

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Vol. 121, Iss. 8 — 24 August 2018

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Images

Figure 1
(a) Layout of the qubits and two-qubit gates (arrows) in the IBM quantum computer ibmqx5. Blue and red qubits were used to measure entanglement spectra and formed subsystem $A$ and $B$ of the simulated quantum spin chain (with periodic boundary conditions), respectively. (b)–(d) Circuits used for constructing the quantum states, where $H$ stands for a Hadamard and $+$ for a cnot gate, respectively. (b) Constructs the ground state of the trivial paramagnet. (c) Constructs the eight qubit cat state (also “GHZ state”). (d) Constructs the ground state of a topological paramagnet (also “graph state”). Hadamard gates that appear next to each other are automatically removed. (e) Symbols for the various logic gates.
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Figure 2
Entanglement spectra measured on the IBM quantum computer (red) compared to the theoretical expectations. Light blue symbols are the theoretical expectations for the respective ideal quantum state. The blue symbols include the effects of sampling noise, present even for an ideal quantum state, ideal gates, and no readout error. Both for the simulated and for the actual measurement, each observable was measured 1024 times. The plots show the eigenvalues of the reduced density matrix: the largest eigenvalues correspond to the lowest levels in the entanglement spectrum. (a) Ground state of a trivial paramagnet: A unique lowest state is found in the entanglement spectrum, reflecting the absence of any entanglement between regions $A$ and $B$ . (b) Eight qubit cat state: A doubly degenerate lowest entanglement level is found reflecting the long-range correlations in this state. (c) Ground state of a topological paramagnet: An entanglement gap separating four low-lying levels is the fingerprint of this topologically nontrivial state. The theoretically expected perfect degeneracy is already substantially lifted by the statistical noise (blue), and in the actual experiment further compromised by decoherence, gate errors, and readout errors. The right-hand panels are eigenvalues of reduced density matrices drawn from the statistical distribution of them when accounting for the statistical noise associated with the measurement.
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Figure 3
(a) Measured mutual information between two regions of the 8-site spin chain. The results are obtained from the measured reduced density matrices, measured in the same way as for Fig. 2. The theoretically exact values are $I (A, B) = 0$ and $I (A, B) = \log 2$ for all $d$ in the case of $| PM ⟩$ and $| cat ⟩$ , respectively, and $I (A, B) = 2 \log 2, \log 2, 0$ for $| SPT ⟩$ at distances $d = 0$ , 1, 2, respectively. (b) Schematic depiction of subsystem $A$ (blue) and subsystem $B$ (red) for the various distances $d = 0$ , 1, 2.
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