Universal photonic quantum computation via time-delayed feedback Hannes Pichlera,b,1,2, Soonwon Choib,1, Peter Zollerc,d, and Mikhail D. Lukinb aInstitute for Theoretical Atomic, Molecular and Optical Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138; bDepartment of Physics, Harvard University, Cambridge, MA 02138; cInstitute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria; dInstitute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, A-6020 Innsbruck, Austria Edited by Harry J. Kimble, California Institute of Technology, Pasadena, CA, and approved September 19, 2017 (received for review June 18, 2017) We propose and analyze a deterministic protocol to generate two- trast, time-delayed quantum feedback renders the system non- dimensional photonic cluster states using a single quantum emit- Markovian, introducing an effective quantum memory that we ter via time-delayed quantum feedback. As a physical implemen- harness to create a universal resource. While previous works tation, we consider a single atom or atom-like system coupled proposed compensating for this limitation of Markovian systems to a 1D waveguide with a distant mirror, where guided photons by using multiple quantum emitters (22), we stress that delayed represent the qubits, while the mirror allows the implementation feedback allows for enabling photonic MBQC already with a of feedback. We identify the class of many-body quantum states single emitter. that can be produced using this approach and characterize them in terms of 2D tensor network states. Photonic 2D Cluster State The fundamental building block of our approach is a single, quantum optics j delayed feedback j photonic quantum computation driven quantum emitter (Q) coupled to a 1D waveguide and a distant mirror as depicted in Fig. 1A. Our protocol consists of a repeated excitation of this quantum emitter, leading to an emis- uantum information processing with optical photons is sion of a train of photon pulses into the waveguide, encoding being actively explored for the past two decades (1, 2). PHYSICS Q qubits via the absence (j0ik ) or presence (j1ik ) of a photon in However, despite a number of conceptual (3) and technolog- the k-th pulse. The mirror feeds these photons back to the emit- ical breakthroughs (4–6), the probabilistic nature of quantum ter with a time delay τ, such that the photons can interact with gates limits the scalability of linear optical systems. Here we the emitter more than once (Fig. 1B). In this way, the emitter can show that one can deterministically generate photonic states with create correlations not only between subsequently emitted pho- the full power of universal quantum computation using quan- tons but also between photon pulses separated by the time delay tum control of a single emitter in combination with time-delayed τ. Effectively this leads to a two-dimensional entanglement struc- coherent quantum feedback (7–9). Our approach is motivated ture as we discuss now for the specific example of the 2D cluster by recent experimental progress demonstrating high-fidelity gen- state (see also Fig. 1C). eration of single photons (5, 10) and deterministic quantum For concreteness, we focus on an emitter (representing Q) operations between single photons and emitters (11–13) in var- with an internal structure depicted in Fig. 2A, supporting two ious systems (14–19). We present an explicit protocol to cre- metastable states jg1i ≡ j0iQ and jg2i ≡ j1iQ, which can be ate a 2D cluster state (20) with a single atomic or atom-like coherently manipulated by a classical field Ω1(t). The emitter emitter coupled to photonic waveguide. The delayed feedback can be excited from jg2i to a state jeLi using a laser with Rabi fre- is introduced by reflecting part of the emitted light field back quency Ω2(t). Following each excitation, the atom will decay to onto the emitter. Our approach allows for deterministic genera- tion of complex photonic entangled states and opens avenues to photonic quantum computation and quantum simulation using Significance minimal resources already available in current state-of-the-art experiments. Creating large entangled states with photons as quantum Time-delayed feedback is a powerful tool in several areas information carriers is a central challenge for quantum infor- of science and engineering (7). In what follows, we show that mation processing. Since photons do not interact directly, delayed coherent quantum feedback can be used as a resource entangling them requires a nonlinear element. One approach to generate quantum entanglement and already in its most basic is to sequentially generate photons using a quantum emit- form may enable universal quantum computation. We focus ter that can induce quantum correlations between photons. on quantum optical systems, where it was shown previously Here we show that delayed quantum feedback dramatically that multipartite entanglement between photonic qubits can be expands the class of achievable photonic quantum states. generated by a single emitter in a sequential emission pro- In particular, we show that in state-of-the-art experiments cess (21–23). We demonstrate that the entanglement structure with single atom-like quantum emitters, the most basic form of the resultant state can be qualitatively enriched using time- of delayed quantum feedback already allows for creation of delayed quantum feedback. While photons emitted in a generic states that are universal resources for quantum computation. sequential process are entangled only in a one-dimensional This opens avenues for quantum information processing with way, characterized by so-called matrix product states (MPSs) photons using minimal experimental resources. (24), we show below that delayed feedback leads to a higher dimensional entanglement structure captured by so-called pro- Author contributions: H.P., S.C., P.Z., and M.D.L. designed research; H.P. and S.C. jected entangled pair states (PEPSs) (25). Significantly, while performed research; H.P. and S.C. analyzed data; and H.P., S.C., P.Z., and M.D.L. wrote the paper. MPSs have limited use for quantum computation and simula- tion, as they can be efficiently simulated classically, PEPSs con- The authors declare no conflict of interest. tain states that serve as a resource for universal measurement- This article is a PNAS Direct Submission. based quantum computation (MBQC) (20). The difference is Published under the PNAS license. rooted in the Markovian nature of the sequential emission pro- 1H.P and S.C. contributed equally to this work. cess that severely restricts the class of achievable states. In con- 2To whom correspondence should be addressed. Email: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1711003114 PNAS Early Edition j 1 of 6 Downloaded by guest on September 28, 2021 A quantum in state jg1i or the photon mode is empty, there is no interaction. emitter This process implements a controlled σz gate mirror 1D wave guide ^ 1 z ZQ;k = j0iQh0j ⊗ k + j1iQh1j ⊗ σk [1] B vaccuum input entangling the atom and the k-th photon. In turn, the subse- quently generated k + N -th photon inherits this entanglement, thereby giving rise to an effective 2D entanglement structure (Fig. 1C). delay line, time delay output: 2D cluster state Formally, the protocol can be interpreted as a sequen- tial application of gates X^Q;k+N Z^Q;k H^Q, on the atom and C photonic qubits k and k + N , on the initial trivial state j0i N j0i C D H^ = p1 (σz + σx ) Q k k (Fig. 2 and ). Here Q 2 Q Q and ^ 1 x XQ;k = j0iQh0|⊗ k + j1iQh1j ⊗ σk . One can show that after (M + 1) × N turns, this gives exactly the 2D cluster state on a M × N square lattice with shifted periodic boundary conditions (see Materials and Methods): 0N (M +1) 1 Y ^ ^ ^ O j C2D i = @ XQ;k+N ZQ;k HQA j0iQ j0ik : [2] k=1 k 2D cluster state Universal quantum computation (27) can be performed by sequentially measuring each photonic qubit directly at the output Fig. 1. Schematic setting. (A) A quantum emitter Q is coupled to a 1D port—for example, using another atom as a high-fidelity mea- waveguide that is terminated on one side by a (distant) mirror. (B) In each surement device. This can be implemented using pulse shaping time step k, Q can emit a phtoton (k + N) toward the mirror—that is, into techniques (Fig. 3A), allowing for a perfect absorption of each a delay line—and interact with a photon k returning from the delay line, photon by the second atom (28, 29). exiting at the output port. (C) Visualization of the resulting entanglement structure: Wrapping the photon channel displayed in B around a cylinder Experimental Requirements. Photons generated in this pulsed with the proper circumference (equivalent to the time delay τ), it can be scheme have a finite bandwidth B. To realize the controlled phase seen that in each time step Q interacts with two photons that are neigh- gate given in Eq. 1, this bandwidth must be small—that is, B γR boring along the artificial, second dimension. Therefore, as time progresses, (26); otherwise, the wave packet is distorted when scattered Q creates entanglement between neighboring photonic qubits, both in the physical and the artificial dimension. by the atom, reducing the gate fidelity, FZ (Fig. 3C). Narrow- bandwidth photons and high-fidelity gates can be obtained by shaping the temporal profiles, eliminating the error to first order jg2i, emitting a photon into the waveguide. For now, we assume that the atom–photon coupling is chiral (26) such that every such photon is emitted unidirectionally—that is, to the left in Fig. 1A AC (see however below). Finally, another excited state jeRi, degen- erate with jeLi, couples to the right, moving photons reflected from the mirror.