Quantum circuit complexity of one-dimensional topological phases

Yichen Huang (黄溢辰) and Xie Chen

Phys. Rev. B 91, 195143 – Published 26 May 2015

Abstract

Topological quantum states cannot be created from product states with local quantum circuits of constant depth and are in this sense more entangled than topologically trivial states, but how entangled are they? Here we quantify the entanglement in one-dimensional topological states by showing that local quantum circuits of linear depth are necessary to generate them from product states. We establish this linear lower bound for both bosonic and fermionic one-dimensional topological phases and use symmetric circuits for phases with symmetry. We also show that the linear lower bound can be saturated by explicitly constructing circuits generating these topological states. The same results hold for local quantum circuits connecting topological states in different phases.

Received 5 August 2014
Revised 8 April 2015

DOI:https://doi.org/10.1103/PhysRevB.91.195143

Authors & Affiliations

Yichen Huang (黄溢辰)^1,* and Xie Chen^1,2

¹Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
²Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

^*yichenhuang@berkeley.edu

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Vol. 91, Iss. 19 — 15 May 2015

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Images

Figure 1
The renormalization group (RG) fixed-point states [15, 22] in the (a) trivial and (b) nontrivial fermionic (Majorana chain) or SPT (e.g., Haldane chain) phases. For states in fermionic phases, each dot represents a Majorana mode and connected pairs form fermionic modes which are vacant or occupied. For states in SPT phases, each dot carries a projective representation of the symmetry group and connected pairs form symmetric singlets. (c) The states in (a) and (b) can be exactly mapped to each other with a linear-depth 2-local quantum circuit composed of swap gates.
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Figure 2
The expectation value $〈 ψ | C^{†} Q C | ψ 〉$ . The horizontal lines attached with small blue squares represent $〈 ψ |$ (bra) or $| ψ 〉$ (ket), and the short rectangles are the 2-local unitaries in $C$ . The (white) unitaries outside the causal cones (dotted lines) of $S^{y}$ (small open red squares) can be removed, as they are symmetric. Then we merge the (gray) symmetric local quantum gates inside each casual cone into one symmetric quantum gate (long rectangle) of sublinear support.
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Figure 3
(a) Graphical representation of MPS (B4) [37]. Each square represents a tensor $A^{(k)}$ with two bond indices (horizontal lines) and one physical index (vertical line). The bond indices are contracted sequentially with periodic boundary conditions (not shown). (b) The condition (B6) for short-range correlated MPS. The graphical equation is approximate up to error $e^{- Ω (k - j)}$ , which can be neglected in the thermodynamic limit $N \to + \infty$ if $k - j = Θ (N)$ . (c) Graphical representation of (B8). The site labels are not shown. (d) is a consequence of (b) and (c). Note that a prefactor of the second, third, and fourth tensor networks is not shown.
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Figure 4
Graphical proof of $〈 ψ | Q^{'} | ψ 〉 = 0$ in the thermodynamic limit $N \to + \infty$ under the assumption that $| ψ 〉$ is in the trivial phase.
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Figure 5
(a) The domain wall (dashed line) that contributes the local phase factor $V (g_{1}) V (g_{2})$ [9]. (b) The short rectangles are the local unitaries in $C$ . The (white) unitaries outside the causal cones (dotted lines) of the domain walls can be removed, as they are symmetric. Then we merge the (gray) symmetric local quantum gates inside each casual cone into one symmetric quantum gate (long rectangle) of sublinear support. (c) Graphical proof of the invariance of the local phase factor for the domain wall in (a) under symmetric local quantum circuits of sublinear depth.
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