Universal quantum computation with temporal-mode bilayer square lattices

Rafael N. Alexander, Shota Yokoyama, Akira Furusawa, and Nicolas C. Menicucci

Phys. Rev. A 97, 032302 – Published 5 March 2018

Abstract

We propose an experimental design for universal continuous-variable quantum computation that incorporates recent innovations in linear-optics-based continuous-variable cluster state generation and cubic-phase gate teleportation. The first ingredient is a protocol for generating the bilayer-square-lattice cluster state (a universal resource state) with temporal modes of light. With this state, measurement-based implementation of Gaussian unitary gates requires only homodyne detection. Second, we describe a measurement device that implements an adaptive cubic-phase gate, up to a random phase-space displacement. It requires a two-step sequence of homodyne measurements and consumes a (non-Gaussian) cubic-phase state.

Received 23 November 2017

DOI:https://doi.org/10.1103/PhysRevA.97.032302

Physics Subject Headings (PhySH)

Quantum information processing with continuous variables

Quantum Information, Science & Technology

Authors & Affiliations

Rafael N. Alexander^1,2,3,*, Shota Yokoyama⁴, Akira Furusawa⁵, and Nicolas C. Menicucci^3,†

¹Center for Quantum Information and Control, University of New Mexico, MSC07-4220, Albuquerque, New Mexico 87131-0001, USA
²Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA
³Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, Victoria 3001, Australia
⁴Centre of Quantum Computation and Communication Technology, School of Engineering and Communication Technology, University of New South Wales, Canberra, Australian Capital Territory 2600, Australia
⁵Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

^*rafaelalexander@unm.edu
^†ncmenicucci@gmail.com

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Vol. 97, Iss. 3 — March 2018

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Figure 1
(a) Optical circuit diagram that executes all stages of the generation and measurement of the bilayer square lattice using temporal modes. Input states encoded within the cluster state can be addressed at the black star ( $★$ ). The legend includes two examples: (left) a switching device for encoding or removing inputs from the cluster state and (right) a syndrome-measurement circuit for quantum error correction using Gottesman-Kitaev-Preskill qubits [36]. The first dashed red line indicates the point at which the state is made up of disjoint squares [see panel (b)]. The second dashed red line is the point where the final cluster state exists [see panel (c)]. Cluster-state modes are measured at detectors $a$ , $b$ , $c$ , or $x$ . Detector $x$ has two settings: homodyne measurement by using $d$ or cubic-phase gate-teleportation measurement by using $e$ and $f$ . The latter involves injecting a cubic-phase-state ancilla $| ϕ_{χ} 〉$ and using an adaptive (variable) phase delay. See main text for details. If the $x$ detector is always set to $d$ , then the phase delays marked by a red asterisk can be omitted by compensating with a $π / 4$ phase delay before detectors $a$ , $b$ , $c$ , and $x$ . (b) Each gray or green rectangle contains modes that arrive at the first red dotted line in panel (a) simultaneously. The numerical labels indicate the relevant time step in units of $Δ t$ . The red arrows between each disjoint square represent the application of the beam splitters between the red dotted lines in panel (a). (c) The bilayer-square-lattice graph. The letters ${a, b, c, x}$ in the rectangle labeled 1 indicate at which detector each mode is measured.
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Figure 2
(a) Measuring $\hat{q} [{(- 1)}^{ξ} \frac{π}{4}]$ at detectors $b$ and $c$ (indicated by a red X) at time index $ξ$ mod $N$ deletes the grayed-out edges [14]. This decouples the central three rows of nodes from the rest of the graph. An input state encoded within the green site on the left will propagate in the direction of the green arrows. (b) After deletion as in panel (a), the middle three graph modes are equivalent to the wire graph shown at the top. This state is known as the dual-rail wire [5, 30]. The letters $a$ , $b$ , $c$ , and $x$ indicate the detector at which the relevant nodes are measured in Fig. 1. Measurements at detectors $x$ and $a$ can be chosen freely to implement gates, whereas measurement settings at $b$ and $c$ are fixed by the deletion measurements.
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Figure 3
(a) Commutation of the ${\hat{C}}_{Z}$ gate through $\hat{B}$ [Eq. (21)]. (b) Commutation of the ${\hat{C}}_{Z}$ gate through ${\hat{B}}^{†}$ [Eq. (21)]. (c) $\hat{q}$ measurement after the ${\hat{C}}_{Z}$ gate [Eq. (22)]. (d) Circuit implementation of the $E$ operation [Eq. (24)]. (e) Measurement-based teleportation identity [Eq. (25)].
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Figure 4
(a) This circuit is equivalent to performing non-Gaussian measurement on the BSL. For compactness, the conditional phase delay shown in Fig. 1 has been incorporated into the homodyne detector at $e$ . The input $| ψ 〉$ should be interpreted as occupying the symmetric subspace of modes $x$ and $a$ [Eq. (19)]. Left of the dotted blue line, this circuit is equivalent to the macronode measurement circuit for a dual-rail wire [30]. The parameter $t_{r} = tanh 2 r$ sets the strength of the ${\hat{C}}_{Z} (t_{r})$ gate. (b) Starting from panel (a), commute the ${\hat{C}}_{Z}$ gate rightwards past one beam splitter [using Fig. 3] and then past the other [using Fig. 3]. The end result is three ${\hat{C}}_{Z}$ gates as shown. (c) Two of these ${\hat{C}}_{Z}$ gates precede a measurement in the $\hat{q}$ basis. We can use Fig. 3 to replace these with $\hat{Z}$ gates. (d) The green box is the measurement-based teleportation circuit [Eq. (25)]. (e) Final equivalent circuit, which implements $\hat{L} (χ, σ, m)$ [Eq. (26)].
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Figure 5
(a) Sublattice containing a region used to implement an entangling gate by measurement-based quantum computation. Detectors $b$ and $c$ measure $\hat{q} [{(- 1)}^{n} \frac{π}{4}]$ at sites marked with a red X, thereby deleting the grayed-out links. Measurement of macronodes 2, 3, and 4 is described in the main text. The measurements on all other modes can be chosen freely. (b) After the deletion in panel (a), the middle six rows of modes are equivalent to this subgraph. Measurements on this resource state can implement circuits as shown. Here, the ${\hat{C}}_{Z} (g)$ gate has weight $g = 2 cot ϕ$ . Also, $ω = {(- 1)}^{k + 1} \frac{3 π}{4}$ and $μ = {(- 1)}^{k} \frac{π}{4}$ .
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