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# Universal photonic quantum computation via time-delayed feedback

Edited by Harry J. Kimble, California Institute of Technology, Pasadena, CA, and approved September 19, 2017 (received for review June 18, 2017)

## Significance

Creating large entangled states with photons as quantum information carriers is a central challenge for quantum information processing. Since photons do not interact directly, entangling them requires a nonlinear element. One approach is to sequentially generate photons using a quantum emitter that can induce quantum correlations between photons. Here we show that delayed quantum feedback dramatically expands the class of achievable photonic quantum states. In particular, we show that in state-of-the-art experiments with single atom-like quantum emitters, the most basic form of delayed quantum feedback already allows for creation of states that are universal resources for quantum computation. This opens avenues for quantum information processing with photons using minimal experimental resources.

## Abstract

We propose and analyze a deterministic protocol to generate two-dimensional photonic cluster states using a single quantum emitter via time-delayed quantum feedback. As a physical implementation, we consider a single atom or atom-like system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits, while the mirror allows the implementation of feedback. We identify the class of many-body quantum states that can be produced using this approach and characterize them in terms of 2D tensor network states.

Quantum information processing with optical photons is being actively explored for the past two decades (1, 2). However, despite a number of conceptual (3) and technological breakthroughs (4⇓–6), the probabilistic nature of quantum gates limits the scalability of linear optical systems. Here we show that one can deterministically generate photonic states with the full power of universal quantum computation using quantum control of a single emitter in combination with time-delayed coherent quantum feedback (7⇓–9). Our approach is motivated by recent experimental progress demonstrating high-fidelity generation of single photons (5, 10) and deterministic quantum operations between single photons and emitters (11⇓–13) in various systems (14⇓⇓⇓⇓–19). We present an explicit protocol to create a 2D cluster state (20) with a single atomic or atom-like emitter coupled to photonic waveguide. The delayed feedback is introduced by reflecting part of the emitted light field back onto the emitter. Our approach allows for deterministic generation of complex photonic entangled states and opens avenues to photonic quantum computation and quantum simulation using minimal resources already available in current state-of-the-art experiments.

Time-delayed feedback is a powerful tool in several areas of science and engineering (7). In what follows, we show that delayed coherent quantum feedback can be used as a resource to generate quantum entanglement and already in its most basic form may enable universal quantum computation. We focus on quantum optical systems, where it was shown previously that multipartite entanglement between photonic qubits can be generated by a single emitter in a sequential emission process (21⇓–23). We demonstrate that the entanglement structure of the resultant state can be qualitatively enriched using time-delayed quantum feedback. While photons emitted in a generic sequential process are entangled only in a one-dimensional way, characterized by so-called matrix product states (MPSs) (24), we show below that delayed feedback leads to a higher dimensional entanglement structure captured by so-called projected entangled pair states (PEPSs) (25). Significantly, while MPSs have limited use for quantum computation and simulation, as they can be efficiently simulated classically, PEPSs contain states that serve as a resource for universal measurement-based quantum computation (MBQC) (20). The difference is rooted in the Markovian nature of the sequential emission process that severely restricts the class of achievable states. In contrast, time-delayed quantum feedback renders the system non-Markovian, introducing an effective quantum memory that we harness to create a universal resource. While previous works proposed compensating for this limitation of Markovian systems by using multiple quantum emitters (22), we stress that delayed feedback allows for enabling photonic MBQC already with a single emitter.

## Photonic 2D Cluster State

The fundamental building block of our approach is a single, driven quantum emitter (Q) coupled to a 1D waveguide and a distant mirror as depicted in Fig. 1*A*. Our protocol consists of a repeated excitation of this quantum emitter, leading to an emission of a train of photon pulses into the waveguide, encoding qubits via the absence (*B*). In this way, the emitter can create correlations not only between subsequently emitted photons but also between photon pulses separated by the time delay τ. Effectively this leads to a two-dimensional entanglement structure as we discuss now for the specific example of the 2D cluster state (see also Fig. 1*C*).

For concreteness, we focus on an emitter (representing Q) with an internal structure depicted in Fig. 2*A*, supporting two metastable states *A* (see however below). Finally, another excited state

### Protocol.

Our protocol starts by first generating 1D cluster states of left-propagating photons (21). To this end, the atom is initially prepared in the state *A* and *B*). The subsequent decay from state

Remarkably, the 2D cluster state is generated from exactly the same sequence if we take into account the effect of the mirror and the scattering of the reflected photons from the atom. We are interested in the situation where the time delay τ is so large that the k-th photon interacts for the second time with the atom between the generation of the *B*). Crucially, if the atom is then in the state *C*).

Formally, the protocol can be interpreted as a sequential application of gates *C* and *D*). Here *Materials and Methods*):*A*), allowing for a perfect absorption of each photon by the second atom (28, 29).

### Experimental Requirements.

Photons generated in this pulsed scheme have a finite bandwidth B. To realize the controlled phase gate given in Eq. **1**, this bandwidth must be small—that is, *C*). Narrow-bandwidth photons and high-fidelity gates can be obtained by shaping the temporal profiles, eliminating the error to first order in *C* and *Materials and Methods*). We note that dispersive distortions due to photon propagation can be absorbed in a redefinition of the qubit states and do not affect the resulting state.

Second, the information capacity of the feedback loop is bounded by the number of photons in the delay line, N. To well distinguish two consecutive photons, one can only generate them at a rate *D*). Therefore, the requirement for narrow-bandwidth photons is competing with the effective size of the emergent, second dimension N. This gives rise to the hierarchy

Apart from these fundamental considerations, experimental imperfections will eventually limit the achievable size of the cluster state. One of the most important challenges is photon loss (33) in atom–photon interactions, often quantified by the so-called cooperativity *E*). High cooperativities have been demonstrated in nanophotonic experiments with neutral atoms and solid-state emitters (13, 15).

Finally, we note that chiral coupling is not essential in realizing the above protocol. For example, in a cavity QED setting (12, 13), the delayed feedback can be introduced by a distant, switchable mirror (Fig. 3*B*). There, proper control of the mirror can ensure that each generated photon interacts exactly twice with the emitter. Moreover, in such a setting one can encode qubit states in photon polarizations rather than number of degrees of freedom, allowing the detection of photon loss errors.

## Theory

In this section, we discuss the generation of 2D entangled photonic states using delayed quantum feedback from a more general perspective. We still consider the paradigmatic setup of delayed quantum feedback in Fig. 1*A* but now study arbitrary protocols that can be realized in such a setting. We completely characterize the class of photonic quantum states that can be generated in this setup in terms of PEPSs. This establishes a remarkable connection between non-Markovian quantum optical systems and condensed matter theory, where PEPSs appear naturally in the description of correlated many-body quantum systems in higher dimensions (25).

To this end, we turn to a more abstract description of the dynamics depicted in Fig. 1 (see also *Materials and Methods*). Formally, the quantum system Q generates an output state by sequential unitary interaction with two qubits k and *B* (see also *Materials and Methods*). Generating specific 2D tensor network states of photons can thus be achieved by proper design of the protocol, corresponding to the tensors **5** can be constructed in this setting with a single quantum emitter and delayed feedback. Per construction, we showed that this includes universal resources for MBQC, but it also allows for designing protocols to create exotic 2D states of light, exhibiting topological order such as string-net states (34) or the ground state of Kitaev’s toric code Hamiltonian (35) (see *Materials and Methods*).

We finally note that this possibility of creating PEPSs as output states of delayed feedback schemes allows for developing variational photonic quantum simulators of strongly correlated 2D systems, extending recently implemented proposals for 1D systems (18, 36) to higher dimensions. This is particularly important, since unlike in 1D, quantum simulators in higher dimensions cannot be efficiently simulated by classical computers.

## Outlook

Our work can be extended in several ways, including the addition of multiple delay lines, leading to tensor networks in higher dimensions. This is of relevance for fault-tolerant implementations of MBQC using 3D cluster states (20). Additionally, on can envisage implementing variants of MBQC that can tolerate up to 50% of counterfactual errors due to photon loss (37). Finally, besides nanophotonic setups, our “single-atom quantum computer” can be implemented in a circuit QED system with microwave photons (16, 18), surface-acoustic, or bulk-acoustic waves (32, 38).

## Materials and Methods

### 2D Cluster State Representation.

In this section, we prove that **3** represents the 2D cluster state. To this end, we start first with the representation of the 1D cluster state on

We now proceed to the construction of the 2D cluster state. From its definition, the 2D cluster state can be obtained from the 1D cluster state above by introducing additional entanglement (via phase gates) between qubits k and **3**.

### General Entanglement Structure.

In this section, we provide details on the explicit connection between the class of achievable states from using a single emitter with delayed quantum feedback and 2D tensor network states (PEPS). To this end, we use induction to show that the repeated application of **4** results in a 2D tensor network that “grows” by an additional tensor in each time step (Fig. 4*B*). This can be seen by induction: we assume that the quantum state of the system after *B*) and *B* with open indices **3** and **4**, new quantum amplitude *A* and by realizing that the summations over *B*). We note that the generated network in Fig. 4*B* is contracted with shifted periodic boundary conditions.

This inductive construction also shows that the arbitrary 2D tensor network satisfying Eq. **5** can be generated from our method via appropriate choice of unitary **5** and can immediately be translated into a protocol similar to the one for generating the 2D cluster state—that is, using only controlled single photon generation and atom–photon phase gates between that atom and feedback photons.

### Imperfections.

Without shaping the wave packets of the emitted photons, each photon produced in a single step has a Lorentzian spectral profile, whose temporal profile is

In the proposed implementation to create the 2D cluster state, the gate **15**—one gets **22** (with

## Acknowledgments

We thank J. I. Cirac, J. Haegeman, L. Jiang, N. Schuch, and F. Verstraete for useful discussions. This work was supported through the National Science Foundation (NSF), the Center for Ultracold Atoms, the Air Force Office of Scientific Research via the Multidisciplinary University Research Initiative, and the Vannevar Bush Faculty Fellowship. H.P. is supported by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory. S.C. acknowledges the support from Kwanjeong Educational Foundation. Work at Innsbruck is supported by the Special Research Program Foundations and Applications of Quantum Science of the Austrian Science Fund, as well as the European Research Council Synergy Grant Ultracold Quantum Matter.

## Footnotes

↵

^{1}H.P and S.C. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: hannes.pichler{at}cfa.harvard.edu.

Author contributions: H.P., S.C., P.Z., and M.D.L. designed research; H.P. and S.C. performed research; H.P. and S.C. analyzed data; and H.P., S.C., P.Z., and M.D.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

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