Fault-Tolerant Quantum Computation with Static Linear Optics

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Fault-Tolerant Quantum Computation with Static Linear Optics

Ilan Tzitrin, Takaya Matsuura, Rafael N. Alexander, Guillaume Dauphinais, J. Eli Bourassa, Krishna K. Sabapathy, Nicolas C. Menicucci, and Ish Dhand

PRX Quantum 2, 040353 – Published 16 December 2021

Abstract

The scalability of photonic implementations of fault-tolerant quantum computing based on Gottesman-Kitaev-Preskill (GKP) qubits is injured by the requirements of inline squeezing and reconfigurability of the linear optical network. In this work we propose a topologically error-corrected architecture that does away with these elements at no cost—in fact, at an advantage—to state preparation overheads. Our computer consists of three modules: a two-dimensional (2D) array of probabilistic sources of GKP states; a depth-four circuit of static beam splitters, phase shifters, and short delay lines; and a 2D array of homodyne detectors. The symmetry of our proposed circuit allows us to combine the effects of finite squeezing and uniform photon loss within the noise model, resulting in more comprehensive threshold estimates. These jumps over both architectural and analytical hurdles considerably expedite the construction of a photonic quantum computer.

Received 9 April 2021
Accepted 15 November 2021

DOI:https://doi.org/10.1103/PRXQuantum.2.040353

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Measurement-based quantum computing Optical quantum information processing Quantum computation Quantum information architectures & platforms Quantum information processing with continuous variables Topological quantum computing

Quantum Information, Science & Technology

Authors & Affiliations

Ilan Tzitrin ^1,2,*,†, Takaya Matsuura^1,3,†, Rafael N. Alexander^1,4,5,†, Guillaume Dauphinais¹, J. Eli Bourassa^1,2, Krishna K. Sabapathy ¹, Nicolas C. Menicucci ^1,4, and Ish Dhand¹

¹Xanadu, Toronto, Ontario M5G 2C8, Canada
²Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
³Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
⁴Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne VIC 3000, Australia
⁵Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131, USA

^*ilan@xanadu.ai
^†These authors contributed equally.

Popular Summary

Photonics is a promising platform for quantum computation in its offer of scalability, mass-manufacturability, largely room-temperature operation, and amenability to various error-correcting codes. Fault-tolerant and universal architectures have been proposed based on optical GKP (Gottesman-Kitaev-Preskill) qubits, which have several favorable properties. However, entangling GKP qubits with each other, as required for universal computation, traditionally involved two demanding kinds of experimental operation: inline squeezing and fast optical switching. In this work we develop an alternative GKP-based architecture that foregoes these requirements while providing additional advantages over existing schemes.

Our proposed architecture works as follows. With some probability, state generation devices prepare slight variations on GKP qubits called sensor states. The devices are multiplexed (that is, they run in parallel) to boost the probability of a successful state generation. The emerging states then proceed through static arrays of beamsplitters that entangle them; it is this stage that is now “passive” (i.e. no longer requires inline squeezing) and “static” (i.e. avoids fast switches). Finally, the states are subject to actively changing measurements, which has the effect of performing the logical computation. We discovered that our passive and static architecture actually lowers the experimental demands on the multiplexing stage of the computation.

There is much work to be done toward a fault-tolerant, scalable, and universal quantum computer, and each platform has a unique set of challenges. Our work improves the prospects and facilitates the development of an optical quantum computer by easing significant hardware challenges.

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Vol. 2, Iss. 4 — December - December 2021

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Figure 1
Panel (a) shows the primal unit cell of the three-dimensional (3D) hybrid pair cluster state, and panels (b) and (c) show the steps for generating it. Panels (b)–(d) are presented as cross sections of waveguide layers stacked in the $Z$ direction, which coincides with the direction of propagation of light through the waveguides. The 3D lattice exists in two spatial ( $X, Y$ ) dimensions and one temporal dimension. The latter is divided into discrete time bins of width $Δ T$ . Colors are included for the relationship between sources and the final state. (b) Waveguide arrangement at the first layer, with each node receiving an input from a source in every $Δ T$ -wide time bin. The time bins for the solid nodes are offset by $Δ T / 2$ , relative to the hollow nodes. As indicated by the arrows, 50:50 beam splitters are applied between pairs of modes, and these generate entangled pairs [see Eq. (7)]. The beam splitters indicated by black arrows create entangled pairs that will connect the state in the $Z$ direction. In (c), crosses indicate the application of a $Δ T / 2$ time-delay line, while slashes indicate the application of a $π / 2$ phase delay. In (d), the state is connected into the macronode cluster state by the application of four additional beam splitters between the four modes that make up each macronode. Dotted beam splitters are applied after solid ones. Note that the time signature of certain nodes changes due to the time-delay lines.
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Figure 2
(a) Circuit representation of the beam splitter network associated with a single macronode, 0, in the case where the central mode is the top wire. Also shown is the connectivity, by beam splitters, to neighboring macronodes. See the legend for circuit conventions. The final four beam splitters correspond to those in Fig. 1. (b) Equivalent circuit to (a), which follows from application of identities (7) and (B14). Here $X_{0}$ denotes displacement $X [(m_{2} + m_{3} - m_{4}) / 2]$ , m₂, m₃, and m₄ denote homodyne measurement outcomes on the satellite modes, as shown in the circuit (B14) and $S$ is the squeezing gate defined in the main text, whose effect is to rescale the homodyne outcomes. (c) Equivalent circuit to (b), which follows from circuit identities that migrate cz gates toward the measurements. The commuted ${CX}^{†}$ gates are depicted with a dotted line because they act trivially on the circuit input (see the main text). The displacements $Z_{1, \dots, 4}$ depend on the measurement outcomes of satellite modes in the neighboring macronodes according to the rules in Appendix 9.
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Figure 3
Swap-out probability $p_{swap}$ over physical error threshold $ϵ$ for the passive architecture. Parameter $ϵ$ combines the effects of finite squeezing (parameter $σ_{fin. sq.}^{2}$ ) and uniform loss (parameter $σ_{loss}^{2} = 1 - η / 2 η$ for transmissivity $η$ ) through $ϵ = σ_{fin. sq.}^{2} + σ_{loss}^{2}$ (i.e., $ϵ$ corresponds to the increase in the variance of the Gaussians to the ideal states’ Wigner functions). Each navy blue point reflects a Monte Carlo threshold search and fit for a given $p_{swap}$ . A minimum-weight-perfect-matching decoder is used with matching graph weights assigned according to estimated qubit-level error probabilities on each node, as in Appendices pp3 and pp4. We find that the error threshold starts at 10.1 dB in the all-GKP case and tends to infinity as the swap-out probability approaches 0.71. This makes our passive and static architecture significantly more tolerant to swap outs compared to the blueprint [6]. Inset: logical failure probability over noise parameter $ϵ$ for an all-GKP macronode RHG code of varying odd distances. We also find an improvement to the no-swap-out threshold compared to Ref. [6].
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Figure 4
Logical failure probability over noise parameter $ϵ$ for a macronode RHG code where each macronode is populated by exactly one GKP state and three momentum-squeezed states. Compare with the infinite squeezing threshold in Fig. 3 at a 75% swap-out rate. With the restriction that each macronode must have a GKP state, no macronode behaves like an effective swap out in the reduced lattice, resulting in a better threshold.
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Figure 5
Graph of the hybrid macronode 3D cluster state. (a) The 2D mode layout, with $Δ T / 2$ offset modes consistent with stages (c) and (D) in Fig. 1—i.e., modes at solid nodes are offset in time by $Δ T / 2$ relative to the modes at hollow nodes. Each macronode consists of four modes, labeled 1–4. (b) The three-dimensional arrangement of the four-mode macronodes is shown. For clarity, the plaques with letters A to F have a color corresponding to a given layer, with lighter colors in deeper layers (green colors for $X$ and $Y$ directions and red for $Z$ ). We omit five modes from the unit cell, corresponding to the back face of the cube. The connectivity in the $X$ - $Y$ plane is identical to the front face. (c) Macronode graph edges for each bond in (b). The top six configurations correspond to weight-1 cz gates, connecting pairs of modes as in Fig. 1. The bottom six configurations correspond to weight- $\pm 1 / 4$ cz gates (the signs are indicated by blue and yellow edge coloring, respectively), showing connectivity of the modes after stage (d) in Fig. 1.
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Figure 6
Schematic of the dependency of byproduct displacements. A group of circles colored in cyan denotes a macronode. A red circle denotes a central mode and a black circle denotes a satellite mode. Modes connected by thin black lines are the initial two-mode cluster states that form the lattice. The solid blue arrow shows the dependency of byproduct displacements in the $q$ quadrature, and a broken blue arrow shows the dependency of byproduct displacements in the $p$ quadrature [see circuits (B18) and (B20)].
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